# Zeta invariants of Morse forms

Jesús A. Álvarez López  Department of Mathematics
University of Santiago de Compostela
15782 Santiago de Compostela
Spain<EMAIL_ADDRESS>,  Yuri A. Kordyukov  Institute of Mathematics
Ufa Federal Research Center
Russian Academy of Sciences
112 Chernyshevsky street
450008 Ufa
Russia<EMAIL_ADDRESS>and  Eric Leichtnam  Institut de Mathématiques
de Jussieu-PRG
CNRS
Batiment Sophie Germain (bureau 740)
Case 7012
75205 Paris Cedex 13, France<EMAIL_ADDRESS>
###### Abstract.

Given a closed real 1-form $\eta$ on a closed Riemannian manifold $(M,g)$, let
$d_{z}$, $\delta_{z}$ and $\Delta_{z}$ be the induced Witten’s type
perturbations of the de Rham derivative and coderivative and the Laplacian on
differential forms on $M$, parametrized by $z\in\mathbb{C}$, and let
$\zeta(s,z)$ be the zeta function of $s\in\mathbb{C}$ given by
$\zeta(s,z)=\operatorname{Tr}^{\text{\rm
s}}({\eta\wedge}\,\delta_{z}\Delta_{z}^{-s})$ when $\Re s\gg 0$. For a class
of Morse forms $\eta$, we prove that $\zeta(s,z)$ is smooth at $s=1$ for $|\Re
z|\gg 0$, and the zeta invariant $\zeta(1,z)$ converges to some
$\mathbf{z}\in\mathbb{R}$ as $\Re z\to+\infty$, uniformly on $\Im z$. We
describe $\mathbf{z}$ in terms of the instantons of an auxiliary Smale
gradient-like vector field $X$ and the Mathai-Quillen current on $TM$ defined
by $g$. Any real cohomology class has a representative $\eta$ satisfying the
needed hypothesis. If $n$ is even, we can prescribe any real value for
$\mathbf{z}$ by perturbing $g$, $\eta$ and $X$; if moreover $M$ is oriented,
we can also achieve the same limit as $\Re z\to-\infty$. This is used to
define and describe certain tempered distributions induced by $g$ and $\eta$.
These distributions appear in another publication as the contributions from
the compact leaves preserved by the flow in a trace formula for simple
foliated flows on closed foliated manifolds, which gives a solution to a
problem proposed by C. Deninger.

###### Key words and phrases:

Witten’s perturbation, Morse form, Morse complex, zeta function of operators,
heat invariant, Ray-Singer metric.

###### 1991 Mathematics Subject Classification:

58A12, 58A14, 58J20, 57R58

The authors are partially supported by the grants MTM2017-89686-P and
PID2020-114474GB-I00 (AEI/FEDER, UE) and ED431C 2019/10 (Xunta de Galicia,
FEDER)

###### Contents

  1. 1 Introduction
  2. 2 Witten’s perturbations
  3. 3 Small and large complexes
  4. 4 Zeta invariants of Morse forms
  5. 5 The small complex vs the Morse complex
  6. 6 Asymptotics of the large zeta invariant
  7. 7 Asymptotics of the small zeta-invariant
  8. 8 Prescription of the asymptotics of the zeta invariant
  9. 9 The switch of the order of integration
  10. A Integrals along instantons

##  1\. Introduction

###  1.1. Witten’s perturbed operators

Let $M$ be a closed $n$-manifold. For any smooth function $h$ on $M$, Witten
[70] introduced a perturbed de Rham differential operator
$d_{\mu}=d+\mu\,{dh\wedge}$, depending on a parameter $\mu\in\mathbb{R}$.
Endowing $M$ with a Riemmanian metric $g$, we have a corresponding perturbed
codifferential operator $\delta_{\mu}=\delta-\mu\,{dh\lrcorner}$, and a
perturbed Laplacian $\Delta_{\mu}=d_{\mu}\delta_{\mu}+\delta_{\mu}d_{\mu}$.
Since $d_{\mu}=e^{-\mu h}\,d\,e^{\mu h}$, it defines the same Betti numbers as
$d$. However $\Delta_{\mu}$ and the usual Laplacian $\Delta$ have different
spectrum in general. In fact, if $h$ is a Morse function and $g$ is Euclidean
with respect to Morse coordinates around the critical points, then the
spectrum of $\Delta_{\mu}$ develops a long gap as $\mu\to+\infty$, giving rise
to the small and large spectrum. The eigenforms of the small/large eigenvalues
generate the small/large subcomplex, $E_{\mu,\text{\rm sm/la}}$. When $h$ is a
Morse function, Witten gave a beautiful analytic proof of the Morse
inequalities by analyzing the small spectrum. This was refined by subsequent
work of Helffer and Sjöstrand [33] and Bismut and Zhang [9, 10], showing that,
if moreover $X:=-\operatorname{grad}h$ is a Smale vector field, then the Morse
complex $(\mathbf{C}^{\bullet},\mathbf{d})$ of $X$ can be considered as the
limit of $(E_{\mu,\text{\rm sm}},d_{\mu})$. More precisely, for certain
perturbed Morse complex $(\mathbf{C}^{\bullet},\mathbf{d}_{\mu})$, isomorphic
to $(\mathbf{C}^{\bullet},\mathbf{d})$, there is a quasi-isomorphism
$\Phi_{\mu}:(E_{z,\text{\rm
sm}},d_{\mu})\to(\mathbf{C}^{\bullet},\mathbf{d}_{\mu})$, defined by
integration on the unstable cells of the zero points of $X$, which becomes an
isomorphism for $\mu\gg 0$ and almost isometric as $\mu\to+\infty$ (after
rescaling at every degree).

We can replace $dh$ with any real closed 1-form $\eta$, obtaining a
generalization of the Witten’s perturbations, $d_{\mu}$, $\delta_{\mu}$ and
$\Delta_{\mu}$. Now $d_{\mu}$ need not be gauge equivalent to $d$, obtaining
new twisted Betti numbers $\beta^{k}_{\mu}$. However the numbers
$\beta^{k}_{\mu}$ have well defined ground values $\beta_{\text{\rm No}}^{k}$,
called the Novikov numbers, which depend upon the de Rham cohomology class
$[\eta]\in H^{1}(M,\mathbb{R})$. Assume that:

  1. (a)

$\eta$ is a Morse form (it has Morse type zeros), and $g$ is Euclidean with
respect to Morse coordinates around the zero points of $\eta$.

(Some concepts used in this section are recalled in Sections 3.1 and 5.1.)
Then $\Delta_{\mu}$ also develops a long gap separating a small spectrum and a
large spectrum, and the analysis of the small spectrum gives Morse
inequalities for the Novikov numbers. Take any auxiliary vector field $X$ such
that:

  1. (b)

$X$ has Morse type zeros, and is gradient-like and Smale; and

  2. (c)

$\eta$ is Lyapunov for $X$, and $\eta$ and $g$ are in standard form with
respect to $X$.

Then the small complex approaches a perturbed Morse complex of $X$. We refer
to work by Novikov [53, 54], Pajitnov [56], Braverman and Farber [12],
Burghelea and Haller [15, 16, 18], and Harvey and Minervini [32, 49].

We can similarly define the perturbation $d_{z}=d+z\,{\eta\wedge}$ with
parameter $z=\mu+i\nu\in\mathbb{C}$ ($\mu,\nu\in\mathbb{R}$ and
$i=\sqrt{-1}$). Its adjoint is $\delta_{z}=\delta-\bar{z}\,{\eta\lrcorner}$,
and we have a corresponding perturbed Laplacian
$\Delta_{z}=d_{z}\delta_{z}+\delta_{z}d_{z}$. As a first step in our study, we
prove extensions of the above results to this case, taking limits as
$|\mu|\to\infty$, uniformly on $\nu$. First, assuming a, we get the long gap
in the spectrum of $\Delta_{z}$ separating the small and large spectrum, which
depends only on $\mu$ (Theorem 3.10). Second, assuming a–c, we show that the
quasi-isomorphism $\Phi_{z}:(E_{z,\text{\rm
sm}},d_{z})\to(\mathbf{C}^{\bullet},\mathbf{d}_{z})$ becomes an isomorphism
for $|\mu|\gg 0$ and almost isometric as $|\mu|\to\infty$ (Theorem 5.3). To
get that the convergence is uniform on $\nu$, we show a version of a Sobolev
inequality for a Sobolev norm defined by $\Delta_{i\nu}$, where the constant
involved is independent of $\nu$ (Proposition 2.2). Then we adapt the
arguments of Bismut and Zhang [9, 10] (see also [71]).

###  1.2. Zeta invariants of some Morse forms

Let $\Pi_{z}^{\perp}$ and $\Pi_{z}^{1}$ be the orthogonal projections to the
images of $\Delta_{z}$ and $d_{z}$, and let $\mathsf{w}$ be the degree
involution. We consider the zeta function
$\zeta(s,z)=\zeta(s,z,\eta):=\zeta(s,\Delta_{z},{\eta\wedge}\,\delta_{z}\mathsf{w})$
[29, Section 1.12.4]. As a function of $s$, this is the meromorphic extension
to $\mathbb{C}$ of the function $\operatorname{Tr}^{\text{\rm
s}}({\eta\wedge}\,\delta_{z}\Delta_{z}^{-s}\Pi_{z}^{\perp})=\operatorname{Tr}^{\text{\rm
s}}({\eta\wedge}\,d_{z}^{-1}\Delta_{z}^{-s+1}\Pi^{1}_{z})$, which is well
defined and holomorphic for $\Re s\gg 0$. Then the zeta invariant $\zeta(1,z)$
would be a renormalized interpretation of the super-trace of
${\eta\wedge}\,d_{z}^{-1}\Pi^{1}_{z}$, which is not of trace class by the
Weyl’s law. However, according to the general theory of zeta functions of
operators, $\zeta(s,z)$ might have a simple pole at $s=1$. To study this zeta
function, we decompose it as sum of the terms defined by the contributions
from the small/large spectrum, $\zeta_{\text{\rm sm/la}}(s,z)=\zeta_{\text{\rm
sm/la}}(s,z,\eta)$. As a function of $s$, $\zeta_{\text{\rm sm}}(s,z)$ is
always holomorphic on $\mathbb{C}$.

For a class of Morse forms, our first main theorem states that $\zeta(s,z)$ is
smooth at $s=1$ for $|\mu|\gg 0$, and describes the asymptotic behavior of
$\zeta(1,z)$ as $\mu\to\pm\infty$, uniformly on $\nu$. In fact, since

$\zeta(s,z,\eta)=-\zeta(s,-z,-\eta)\;,\quad\zeta_{\text{\rm
sm/la}}(s,z,\eta)=-\zeta_{\text{\rm sm/la}}(s,-z,-\eta)\;,$ (1.1)

it is enough to consider the case where $\mu\gg 0$ and take the limit as
$\mu\to+\infty$.

We use the current $\psi(M,\nabla^{M})$ of degree $n-1$ on $TM$ constructed by
Mathai and Quillen in [42], depending on the Levi-Civita connection
$\nabla^{M}$. This current is smooth on the complement of the zero section,
where it is given by the solid angle. It is also locally integrable, and its
wave front set is contained in the conormal bundle in $T^{*}TM$ of the zero
section of $TM$. Since this set does not meet the conormal bundle of the map
$X:M\rightarrow TM$ (assuming b), $X^{*}\psi(M,\nabla^{M})$ is well defined as
a current on $M$. Assuming a–c, consider the real number

$\mathbf{z}_{\text{\rm la}}=\mathbf{z}_{\text{\rm
la}}(M,g,\eta)=\int_{M}\eta\wedge X^{*}\psi(M,\nabla^{M})\;,$

which is known to be independent of $X$ [9, Proposition 6.1].

Now suppose also that:

  1. (d)

for every zero point $p$ of $X$, the maximum value of the integrals of $\eta$
along the instantons of $X$ with $\alpha$-limit $\\{p\\}$ only depends on the
Morse index $k$ of $p$.

This maximum value is denoted by $-a_{k}$ for some $a_{k}>0$. Let $m^{1}_{k}$
denote the dimension of $d_{z}(E_{z,\text{\rm sm}}^{k-1})$ for $|\mu|\gg 0$,
which is independent of $z$. Consider also the real number

$\mathbf{z}_{\text{\rm sm}}=\mathbf{z}_{\text{\rm
sm}}(M,g,\eta,X)=\sum_{k=1}^{n}(-1)^{k}\big{(}1-e^{a_{k}}\big{)}m^{1}_{k}\;,$

and let $\mathbf{z}=\mathbf{z}(M,g,\eta,X)=\mathbf{z}_{\text{\rm
sm}}+\mathbf{z}_{\text{\rm la}}$.

######  Theorem 1.1.

Let $M\equiv(M,g)$ be a closed Riemannian manifold of dimension $n$. Let
$\eta$ be a closed real $1$-form on $M$ satisfying a.

  1. (i)

For $\mu\gg 0$, $\zeta(s,z)$ is smooth for $s\neq(1-k)/2\ (k=0,1,\dots)$, and

$\zeta(1,z)=\lim_{t\downarrow 0}\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,d_{z}^{-1}e^{-t\Delta_{z}}\Pi^{1}_{z}\big{)}\;.$

  2. (ii)

Let $X$ be a vector field on $M$ satisfying b–c. Then

$\zeta_{\text{\rm la}}(1,z)=\mathbf{z}_{\text{\rm la}}+O(\mu^{-1/2})$

as $\mu\to+\infty$, uniformly on $\nu$.

  3. (iii)

If moreover d holds, then

$\zeta_{\text{\rm sm}}(1,z)=\mathbf{z}_{\text{\rm sm}}+O(\mu^{-1})$

as $\mu\to+\infty$, uniformly on $\nu$.

The existence of the limit of Theorem 1.1 i is rather surprising because
${\eta\wedge}\,d_{z}^{-1}e^{-t\Delta_{z}}\Pi_{z}^{1}$ is weakly convergent to
${\eta\wedge}\,d_{z}^{-1}\Pi_{z}^{1}$. An expression similar to
$\operatorname{Tr}^{\text{\rm
s}}({\eta\wedge}\,d_{z}^{-1}e^{-t\Delta_{z}}\Pi_{z}^{1})$ was used by Mrowka,
Ruberman and Saveliev to define a cyclic eta invariant [50]. Theorem 1.1 iii
shows that $\mathbf{z}_{\text{\rm sm}}$ and $\mathbf{z}$ are also independent
of $X$. Thus $X$ will be omitted in their notation. In the notation of
$\mathbf{z}_{\text{\rm sm/la}}$ and $\mathbf{z}$, we may also omit $M$ or $g$
if they are fixed.

By (1.1), if we take $\mu\to-\infty$ in Theorem 1.1, we have to replace
$\mathbf{z}_{\text{\rm sm/la}}(\eta)$ with $-\mathbf{z}_{\text{\rm
sm/la}}(-\eta)$. Descriptions of $-\mathbf{z}_{\text{\rm sm/la}}(-\eta)$ are
given in (6.9) and (7.1).

Our second main theorem is about the prescription of
$\mathbf{z}=\mathbf{z}(M,g,\eta)$ without changing the cohomology class of
$\eta$.

######  Theorem 1.2.

Let $M$ be a smooth closed manifold of dimension $n$.

  1. (i)

Let $X$ be a vector field satisfying b. For every $\xi\in H^{1}(M,\mathbb{R})$
and $\tau\gg 0$, there is some $\eta\in\xi$ and a Riemannian metric $g$
satisfying a, c and d with $X$ such that $\mathbf{z}(M,g,\eta)=\tau$. If $n$
is even, this property holds for all $\tau\in\mathbb{R}$.

  2. (ii)

Assume $M$ is oriented and $n$ is even. Then, for every $\xi\in
H^{1}(M,\mathbb{R})$ and $\tau\in\mathbb{R}$, there is some $\eta\in\xi$, a
Riemannian metric $g$ and a vector field $X$ satisfying a–d such that
$\pm\mathbf{z}(M,g,\pm\eta)=\tau$.

###  1.3. A distribution associated to some Morse forms

A trace formula for simple foliated flows on closed foliated manifolds was
conjectured by C. Deninger (see e.g. [22]). He was motivated by analogies with
a formula in Arithmetics, and previous work of Guillemin and Sternberg [30].
This trace formula is an expression for a Lefschetz distribution in terms of
infinitesimal data of the flow at the fixed points and closed orbits. This
Lefschetz distribution should be an analogue of the Lefschetz number for the
action induced by the flow on some leafwise cohomology, whose value is a
distribution on $\mathbb{R}$—the precise definition of these notions is part
of the problem. In [3, 4], the first two authors proved such a trace formula
when the flow has no preserved leaves; see also the contributions [40, 41] by
the third author. The general case is considerably more involved. In [5], we
propose a solution to this problem using a few additional ingredients. One of
them is the b-trace introduced by Melrose [43]. Since the b-trace is not
really a trace, it produces an extra term, denoted by $Z$, in the same way as
the eta invariant shows up in Index Theory on manifolds with boundary. In our
trace formula, the term $Z$ is a contribution from the compact leaves
preserved by the flow, which depends on the choice of a form defining the
foliation and a metric on the ambient manifold. But $Z$ may not be well
defined in general; it will be proved that appropriate choices of the form and
the metric guarantee its existence.

Precisely, we would like to define

$Z=Z(M,g,\eta)=\lim_{\mu\to+\infty}Z_{\mu}\;,$ (1.2)

in the space of tempered distributions, where $Z_{\mu}=Z_{\mu}(M,g,\eta)$
($\mu\gg 0$) should be a tempered distribution defined by

$\langle
Z_{\mu},f\rangle=-\frac{1}{2\pi}\int_{0}^{\infty}\int_{-\infty}^{\infty}\operatorname{Tr}^{\text{\rm
s}}\left({\eta\wedge}\,\delta_{z}e^{-u\Delta_{z}}\right)\,\hat{f}(\nu)\,d\nu\,du\;,$
(1.3)

for any Schwartz function $f$, where $\operatorname{Tr}^{\text{\rm s}}$
denotes the supertrace and $\hat{f}$ the Fourier transform of $f$.

Let $\delta_{0}$ denote the Dirac distribution at $0$ on $\mathbb{R}$. The
problem about the definition of $Z$ is solved in our third main theorem for
the same class of Morse forms as before.

######  Theorem 1.3.

Let $M\equiv(M,g)$ be a closed Riemannian manifold of dimension $n$. Let
$\eta$ be a closed $1$-form on $M$ satisfying a, c and d with some vector
field satisfying b. Then (1.2) and (1.3) define the tempered distribution
$Z=\mathbf{z}\delta_{0}$.

According to Theorems 1.2 and 1.3, we can choose $\eta$ and $g$ in the trace
formula for foliated flows so that $Z(M,g,\eta)=0$ if $n$ is even, or
$Z(M,g,\pm\eta)=0$ if moreover $M$ is oriented, achieving the original
expression of Deninger’s conjecture.

It looks clear that extensions of Theorems 1.1, 1.2 and 1.3 with coefficients
in flat vector bundles could be similarly proved. We only consider complex
coefficients for the sake of simplicity since this is enough for our
application.

###  1.4. Some ideas of the proofs of Theorems 1.1, 1.2 and 1.3

Theorem 1.1 i follows by using that the derived heat trace invariants up to
order $n$ of the elliptic complex $d_{z}$ are independent of $z$, proved by
Gilkey and the first author [1] (Section 4.6).

Consider the meromorphic function

$\theta(s,z)=-\zeta(s,\Delta_{z},\mathsf{N}\mathsf{w})\;,$ (1.4)

where $\mathsf{N}$ is the number operator, and write

$\theta(s,z)=\theta_{\text{\rm sm}}(s,z)+\theta_{\text{\rm la}}(s,z)\;,\\\ $

where

$\theta_{\text{\rm sm/la}}(s,z)=-\zeta_{\text{\rm
sm/la}}(s,\Delta_{z},\mathsf{N}\mathsf{w})\;,$ (1.5)

using the contributions from the small/large spectrum as above. Thus
$e^{\theta^{\prime}(0,z)/2}$ is the factor used to define the Ray-Singer
metric on $\det H_{z}^{\bullet}(M)$ [9]. We obtain (Corollary 4.16)

$\zeta_{\text{\rm la}}(1,z)=\partial_{z}\theta_{\text{\rm
la}}^{\prime}(0,z)\;,$

using a prime to denote $\partial_{s}$. This equality allows us to use the
deep relation between the Ray-Singer metric and the Morse metric on $\det
H_{z}^{\bullet}(M)$, proved by Bismut-Zhang [9, 10]. In this way, using also
that $\Phi_{z}:E_{z,\text{\rm sm}}\to\mathbf{C}^{\bullet}$ is an isomorphism,
we obtain that $\zeta_{\text{\rm la}}(1,z)$ is asymptotic to
$\mathbf{z}_{\text{\rm la}}$ as $\mu\to+\infty$ (Section 6.2). This proves
Theorem 1.1 ii.

When $\eta$ is exact, we show this asymptotic expression of $\zeta_{\text{\rm
la}}(1,z)$ assuming only a (Section 4.7), without using the indicated strong
result of Bismut-Zhang. Instead, we apply that the index density of the
elliptic complex $d_{z}$ is independent of $z$, proved by Gilkey and the first
author [2] and by the authors [5].

On the other hand, given any $\xi\in H^{1}(M,\mathbb{R})$ and a vector field
$X$ satisfying b, we prove that there is some $\eta\in\xi$ and a metric $g$
satisfying a, c and d (Theorem 7.1). This can be considered as an extension of
a theorem of Smale stating the existence of nice Morse functions [67, Theorem
B] (the case where $\xi=0$). Its proof is relegated to Appendix A because of
its different nature.

The properties a–d are used to give an asymptotic description of
$\mathbf{d}_{z}$ as $\mu\to+\infty$ (Section 7.2). From this asymptotic
description and using that $\Phi_{z}:E_{z,\text{\rm
sm}}\to\mathbf{C}^{\bullet}$ is an isomorphism for $\mu\gg 0$, we get upper
and lower bounds of the nonzero small spectrum of $\Delta_{z}$ (Theorem 7.5),
which are independent of $\nu$. This is a partial extension of accurate
descriptions of the nonzero small eigenvalues achieved in the case where
$\eta$ is exact and the parameter is real [39, 45]. With the same procedure
and using the bounds of the nonzero small spectrum, it also follows that
$\zeta_{\text{\rm sm}}(1,z)$ is asymptotic to $\mathbf{z}_{\text{\rm sm}}$ as
$\mu\to+\infty$ (Section 7.4), showing Theorem 1.1 iii.

Next, by modifying $\eta$ and $X$ around its zero points of index $0$ and $n$,
without changing the cohomology class of $\eta$, we can achieve any real
number as $\mathbf{z}(\eta)$, or as both $\pm\mathbf{z}(\pm\eta)$ if $M$ is
oriented and $n$ even (Section 8). This shows Theorem 1.2.

If it is possible to switch the order of integration in (1.3),

$\displaystyle\langle Z_{\mu},f\rangle$
$\displaystyle=-\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{0}^{\infty}\operatorname{Tr}^{\text{\rm
s}}\left({\eta\wedge}\,\delta_{z}e^{-u\Delta_{z}}\right)\,\hat{f}(\nu)\,du\,d\nu$
$\displaystyle=\frac{1}{2\pi}\int_{-\infty}^{\infty}\lim_{t\downarrow
0}\operatorname{Tr}^{\text{\rm
s}}\left({\eta\wedge}\,d_{z}^{-1}e^{-t\Delta_{z}}\Pi_{z}^{1}\right)\,\hat{f}(\nu)\,d\nu\;,$
(1.6)

then Theorem 1.3 is an easy consequence of Theorem 1.1. Thus it only remains
to prove that both (1.3) and (1.6) define the same tempered distribution
$Z_{\mu}$. This follows from the Lebesgue’s dominated convergence theorem and
Fubini’s theorem (Section 9). The verification of the hypothesis of the
Fubini’s theorem requires the above lower estimate of the nonzero spectrum.

##  2\. Witten’s perturbations

###  2.1. Preliminaries on the Witten’s perturbations

####  2.1.1. Basic notation

Let $M$ be a closed Riemannian $n$-manifold. For any smooth
Euclidean/Hermitean vector bundle $E$ over $M$, let $C^{m}(M;E)$,
$C^{\infty}(M;E)$, $L^{2}(M;E)$, $L^{\infty}(M;E)$ and $H^{m}(M;E)$ denote the
spaces of distributional sections that are $C^{m}$, $C^{\infty}$, $L^{2}$,
$L^{\infty}$ and of Sobolev order $m$, respectively; as usual, $E$ is removed
from this notation if it is the trivial line bundle. Consider the induced
scalar product $\langle\ ,\ \rangle$ and norm $\|\ \|$ on $L^{2}(M;E)$, and
the induced norm $\|\ \|_{L^{\infty}}$ on $L^{\infty}(M;E)$. Fix also norms,
$\|\ \|_{m}$ on every $H^{m}(M;E)$ and $\|\ \|_{C^{m}}$ on $C^{m}(M;E)$ with
$\|\ \|=\|\ \|_{0}$ and $\|\ \|_{C^{0}}=\|\ \|_{L^{\infty}}$. If $P$ is the
orthogonal projection of $L^{2}(M;E)$ to some closed subspace $V$, then
$P^{\perp}$ denotes the orthogonal projection to $V^{\perp}$.

Let $T_{\mathbb{C}}M=TM\otimes\mathbb{C}$ and
$T_{\mathbb{C}}^{*}M=T^{*}M\otimes\mathbb{C}$. The exterior bundle with
coefficients in $\mathbb{K}=\mathbb{R},\mathbb{C}$ is denoted by
$\Lambda_{\mathbb{K}}=\Lambda_{\mathbb{K}}M$, and let
$\Omega(M,\mathbb{K})=C^{\infty}(M;\Lambda_{\mathbb{K}})$; in particular,
$C^{\infty}(M,\mathbb{K})=\Omega^{0}(M,\mathbb{K})$. The Levi-Civita
connection is denoted by $\nabla=\nabla^{M}$. As usual, $d$ and $\delta$
denote the de Rham derivative and coderivative, and let $D=d+\delta$ and
$\Delta=D^{2}=d\delta+\delta d$ (the Laplacian). Let $Z(M,\mathbb{K})$ and
$B(M,\mathbb{K})$ denote the kernel and image of $d$ in
$\Omega(M,\mathbb{K})$. Thus
$H^{\bullet}(M,\mathbb{K})=Z(M,\mathbb{K})/B(M,\mathbb{K})$ is the de Rham
cohomology with coefficients in $\mathbb{K}$. We typically consider complex
coefficients, so we will omit $\mathbb{K}$ from all of the above notation just
when $\mathbb{K}=\mathbb{C}$. Take $\|\ \|_{m}$ and $\|\ \|_{C^{m}}$ given on
$\Omega(M)$ by

$\|\alpha\|_{m}=\sum_{k=0}^{m}\|D^{k}\alpha\|\;,\quad\|\alpha\|_{C^{m}}=\sum_{k=0}^{m}\|\nabla^{k}\alpha\|_{L^{\infty}}\;.$

For any homogeneous linear operator between graded vector spaces,
$T:V^{\bullet}\to W^{\bullet}$, the notation $T_{k}$ means its precomposition
with the canonical projection of $V^{\bullet}$ to $V^{k}$. On any graded
vector space $V^{\bullet}$, let $\mathsf{w}$ and $\mathsf{N}$ be the degree
involution and number operator; i.e., $\mathsf{w}=(-1)^{k}$ and $\mathsf{N}=k$
on $V^{k}$.

For any $\eta\in\Omega^{1}(M,\mathbb{R})$ with
$\eta^{\sharp}=X\in\mathfrak{X}(M):=C^{\infty}(M;TM)$, let $\mathcal{L}_{X}$
and $\iota_{X}$ denote the Lie derivative and interior product with respect to
$X$, and let ${\eta\lrcorner}=-(\eta\wedge)^{*}=-\iota_{X}$. Using the
identity $\operatorname{Cl}(T^{*}M)\equiv\Lambda M$ defined by the symbol of
filtered algebras, the left Clifford multiplication by $\eta$ is
$c(\eta)={\eta\wedge}+{\eta\lrcorner}$, and the composition of $\mathsf{w}$
with the right Clifford multiplication by $\eta$ is
$\hat{c}(\eta)={\eta\wedge}-{\eta\lrcorner}$; in particular,
$c(\eta)^{*}=-c(\eta)$ and $\hat{c}(\eta)^{*}=\hat{c}(\eta)$. Recall that, for
any $h\in C^{\infty}(M,\mathbb{R})$, $\alpha\in\Omega(M)$ and
$\eta,\theta\in\Omega^{1}(M,\mathbb{R})$,

$\displaystyle D(h\alpha)=hD\alpha+\hat{c}(dh)\alpha\;,$ (2.1) $\displaystyle
c(\eta)\hat{c}(\theta)+\hat{c}(\theta)c(\eta)=0\;.$ (2.2)

In the whole paper, unless otherwise indicated, we will use the following
notation without further comment. We use constants $C,c>0$ without even
mentioning their existence, and their precise values may change from line to
line. We may add subindices or primes to these constants if needed. We also
use a complex parameter $z=\mu+i\nu\in\mathbb{C}$ ($\mu,\nu\in\mathbb{R}$ and
$i=\sqrt{-1}$). Recall that $\partial_{z}=(\partial_{\mu}-i\partial_{\nu})/2$
and $\partial_{\bar{z}}=(\partial_{\mu}+i\partial_{\nu})/2$.

####  2.1.2. Perturbations defined by a closed real 1-form

For any $\omega\in Z^{1}(M)$, we have the Witten’s type perturbations
$d_{\omega}$, $\delta_{\omega}$, $D_{\omega}$ and $\Delta_{\omega}$ of $d$,
$\delta$, $D$ and $\Delta$. Given $\eta\in Z^{1}(M,\mathbb{R})$ and
$z\in\mathbb{C}$, we write $d_{z}=d_{z\eta}$, $\delta_{z}=\delta_{\eta}$,
$D_{z}=D_{z\eta}$ and $\Delta_{z}=\Delta_{z\eta}$. These operators have the
following expressions:

$\left.\begin{aligned}
d_{z}&=d+z\,\eta\wedge\;,\quad\delta_{z}=d_{z}^{*}=\delta-\bar{z}\,{\eta\lrcorner}\;,\\\
D_{z}&=d_{z}+\delta_{z}=D+\mu\hat{c}(\eta)+i\nu
c(\eta)=D_{i\nu}+\mu\hat{c}(\eta)\;,\\\
\Delta_{z}&=D_{z}^{2}=d_{z}\delta_{z}+\delta_{z}d_{z}\\\
&=\Delta+\mu\mathsf{H}_{\eta}+i\nu\mathsf{J}_{\eta}+|z|^{2}|\eta|^{2}=\Delta_{i\nu}+\mu\mathsf{H}_{\eta}+\mu^{2}|\eta|^{2}\;,\end{aligned}\right\\}$
(2.3)

where, for $X=\eta^{\sharp}$,

$\displaystyle\mathsf{H}_{\eta}$
$\displaystyle=D\hat{c}(\eta)+\hat{c}(\eta)D=\mathcal{L}_{X}^{*}+\mathcal{L}_{X}\;,$
$\displaystyle\mathsf{J}_{\eta}$
$\displaystyle=Dc(\eta)+c(\eta)D=\mathcal{L}_{X}^{*}-\mathcal{L}_{X}\;.$

Note that $\mathsf{H}_{\eta}$ is of order zero and $\mathsf{J}_{\eta}$ of
order one.

As families of operators, $d_{z}$ and $\delta_{z}$ are holomorphic and anti-
holomorphic functions of $z$, respectively. More precisely, it follows from
(2.3) that

$\left.\begin{aligned}
\partial_{z}d_{z}&={\eta\wedge}\;,&\quad\partial_{z}\delta_{z}&=0\;,&\quad\partial_{z}\Delta_{z}&={\eta\wedge}\,\delta_{z}+\delta_{z}\,{\eta\wedge}\;,\\\
\partial_{\bar{z}}d_{z}&=0\;,&\quad\partial_{\bar{z}}\delta_{z}&=-{\eta\lrcorner}\;,&\quad\partial_{\bar{z}}\Delta_{z}&=-{\eta\lrcorner}\,d_{z}-d_{z}\,{\eta\lrcorner}\;.\end{aligned}\right\\}$
(2.4)

The operator $d_{z}$ defines an elliptic complex on $\Omega(M)$, whose
cohomology is denoted by $H_{z}^{\bullet}(M)$. Since $d_{z}$ has the same
principal symbol as $d$, it is a generalized Dirac complex and $\Delta_{z}$ a
generalized Laplacian [6, Definition 2.2]. If $\theta=\eta+dh$ for some $h\in
C^{\infty}(M,\mathbb{R})$, then the multiplication operator

$e^{zh}:(\Omega(M),d_{z\theta})\to(\Omega(M),d_{z\eta})$ (2.5)

is an isomorphism of differential complexes, and therefore it induces an
isomorphism $H_{z\theta}^{\bullet}(M)\cong H_{z\eta}^{\bullet}(M)$. Thus the
isomorphism class of $H_{z}^{\bullet}(M)$ only depends on $\xi:=[\eta]\in
H^{1}(M,\mathbb{R})$ and $z\in\mathbb{C}$. By ellipticity, $D_{z}$ and
$\Delta_{z}$ have a discrete spectrum, and there is a Hodge type decomposition

$\Omega(M)=\ker\Delta_{z}\oplus\operatorname{im}d_{z}\oplus\operatorname{im}\delta_{z}\;,$
(2.6)

as topological vector spaces. It induces a Hodge type isomorphism

$H_{z}^{\bullet}(M)\cong\ker\Delta_{z}\;.$ (2.7)

We also have

$\displaystyle\ker\Delta_{z}$ $\displaystyle=\ker D_{z}=\ker
d_{z}\cap\ker\delta_{z}\;,$ $\displaystyle\operatorname{im}\Delta_{z}$
$\displaystyle=\operatorname{im}D_{z}=\operatorname{im}d_{z}\oplus\operatorname{im}\delta_{z}\;.$

The orthogonal projections of $\Omega(M)$ to $\ker\Delta_{z}$,
$\operatorname{im}d_{z}$ and $\operatorname{im}\delta_{z}$ are denoted by
$\Pi_{z}=\Pi^{0}_{z}$, $\Pi^{1}_{z}$ and $\Pi^{2}_{z}$, respectively; thus
$\Pi_{z}^{\perp}=\Pi^{1}_{z}+\Pi^{2}_{z}$. The restrictions
$d_{z}:\operatorname{im}\delta_{z}\to\operatorname{im}d_{z}$,
$\delta_{z}:\operatorname{im}d_{z}\to\operatorname{im}\delta_{z}$ and
$D_{z}:\operatorname{im}D_{z}\to\operatorname{im}D_{z}$ are topological
isomorphisms, and therefore the compositions $d_{z}^{-1}\Pi^{1}$,
$\delta_{z}^{-1}\Pi^{2}$ and $D_{z}^{-1}\Pi^{\perp}$ are defined and
continuous on $\Omega(M)$. Moreover the diagram

$\begin{CD}\operatorname{im}\delta_{z,k+1}@>{d_{z,k}}>{}>\operatorname{im}d_{z,k}\\\
@V{\Delta_{z,k}}V{}V@V{}V{\Delta_{z,k+1}}V\\\
\operatorname{im}\delta_{z,k+1}@>{d_{z,k}}>{}>\operatorname{im}d_{z,k}\end{CD}$
(2.8)

is commutative—recall that $d_{z,k}$, $\delta_{z,k}$ and $\Delta_{z,k}$ are
the compositions of the projection to $\Omega^{k}(M)$ with $d_{z}$,
$\delta_{z}$ and $\Delta_{z}$. The alternate sum of the dimensions
$\beta_{z}^{k}=\beta_{z}^{k}(M,\xi)=\dim H_{z}^{k}(M)$ ($k=0,\dots,n$) is the
Euler characteristic [26, Proposition 1.40],

$\sum_{k}(-1)^{k}\beta_{z}^{k}=\chi(M)\;.$ (2.9)

(This is also a consequence of the index theorem.) For every degree $k$,
$\beta_{z}^{k}$ is independent of $z$ outside a discrete subset of
$\mathbb{C}$, where $\beta_{z}^{k}$ jumps (Mityagin and Novikov [55, Theorem
1]). This ground value of $\beta_{z}^{k}$ is called the $k$-th _Novikov Betti
number_ , and will be denoted by $\beta_{\text{\rm No}}^{k}=\beta_{\text{\rm
No}}^{k}(M,\xi)$. Moreover it will be shown in Section 5.2.2 that

$\beta_{z}^{k}=\beta_{\text{\rm No}}^{k}\quad\text{for}\quad|\mu|\gg 0\;.$
(2.10)

(When $z$ is real, this is proved in [25, Theorem 2.8], [12, Lemma 1.3], [16,
Proposition 4].) Thus the discrete set of parameters $z\in\mathbb{C}$ with
$\beta_{z}^{k}(M,\xi)>\beta_{\text{\rm No}}^{k}(M,\xi)$ for some degree $k$ is
contained in a strip $|\mu|\leq C$. The notation
$N_{z}=\beta_{z}^{0}+\dots+\beta_{z}^{n}$ and $N_{\text{\rm
No}}=\beta_{\text{\rm No}}^{0}+\dots+\beta_{\text{\rm No}}^{n}$ will be used.

By (2.3) and since $\eta$ is real, for all $\alpha\in\Omega(M)$,

$\overline{d_{z}\alpha}=d_{\bar{z}}\bar{\alpha}\;,\quad\overline{\delta_{z}\alpha}=\delta_{\bar{z}}\bar{\alpha}\;,\quad\overline{D_{z}\alpha}=D_{\bar{z}}\bar{\alpha}\;,\quad\overline{\Delta_{z}\alpha}=\Delta_{\bar{z}}\bar{\alpha}\;.$
(2.11)

So conjugation induces $\mathbb{C}$-antilinear $\mathbb{R}$-isomorphisms

$H_{z}^{k}(M)\cong
H_{\bar{z}}^{k}(M)\;,\quad\ker\Delta_{z,k}\cong\ker\Delta_{\bar{z},k}\;,$

yielding $\beta_{z}^{k}=\beta_{\bar{z}}^{k}$.

####  2.1.3. Case of an exact form

When $\eta=dh$ for some $h\in C^{\infty}(M,\mathbb{R})$, we have the original
Witten’s perturbations, which satisfy

$\left.\begin{gathered}d_{z}=e^{-zh}\,d\,e^{zh}=e^{-i\nu h}\,d_{\mu}\,e^{i\nu
h}\;,\\\ \delta_{z}=e^{\bar{z}h}\,\delta\,e^{-\bar{z}h}=e^{-i\nu
h}\,\delta_{\mu}\,e^{i\nu h}\;,\\\ D_{z}=e^{-i\nu h}\,D_{\mu}\,e^{i\nu
h}\;,\quad\Delta_{z}=e^{-i\nu h}\,\Delta_{\mu}\,e^{i\nu
h}\;.\end{gathered}\right\\}$ (2.12)

Thus the multiplication operator

$e^{zh}:(\Omega(M),d_{z})\to(\Omega(M),d)$ (2.13)

is an isomorphism of differential complexes, and therefore
$H_{z}^{\bullet}(M)\cong H^{\bullet}(M)$; thus
$\beta_{z}^{k}=\beta^{k}=\beta^{k}(M)$ (the $k$th Betti number) in this case.
Moreover multiplication by $e^{i\nu h}$ defines a unitary isomorphism
$\ker\Delta_{z}\cong\ker\Delta_{\mu}$.

####  2.1.4. Interpretation of the closed form as a flat connection

There is a unique flat connection $\nabla^{M\times\mathbb{C}}$ on the trivial
complex line bundle $M\times\mathbb{C}$ so that
$\nabla^{M\times\mathbb{C}}1=\eta$. The corresponding flat complex line bundle
is denoted by $\mathcal{L}=\mathcal{L}_{\eta}$. Note that
$\mathcal{L}_{z\eta}=\mathcal{L}_{\eta}^{z}$. Since every $\mathcal{L}^{z}$ is
canonically trivial as line bundle, it has a canonical Hermitian structure
$g^{\mathcal{L}^{z}}$. Let
$(\Omega(M,\mathcal{L}^{z})=\Omega(M),d^{\mathcal{L}^{z}})$ be the de Rham
complex with coefficients in $\mathcal{L}^{z}$.

Suppose $\eta=dh$ for a while. Then the horizontal leaves of $\mathcal{L}^{z}$
are the graphs of the functions $se^{-zh}$ ($s\in\mathbb{C}$). So
$e^{zh}:\mathcal{L}^{z}\to\mathcal{L}^{0}$ is an isomorphism of flat bundles,
which induces an isomorphism of differential complexes,

$e^{zh}:(\Omega(M),d^{\mathcal{L}^{z}})\to(\Omega(M),d)\;.$

Comparing with the isomorphism (2.13), we get $d^{\mathcal{L}^{z}}=d_{z}$ on
$\Omega(M)$. Furthermore, since $g^{\mathcal{L}^{z}}$ corresponds to $e^{2\mu
h}g^{\mathcal{L}^{0}}$ via the isomorphism
$e^{zh}:\mathcal{L}^{z}\to\mathcal{L}^{0}$, it follows that
$\nabla^{\mathcal{L}^{z}}g^{\mathcal{L}^{z}}$ corresponds to

$\nabla^{\mathcal{L}^{0}}\big{(}e^{2\mu h}g^{\mathcal{L}^{0}}\big{)}=2\mu
e^{2\mu h}\,dh\otimes g^{\mathcal{L}^{0}}\;,$

yielding

$\nabla^{\mathcal{L}^{z}}g^{\mathcal{L}^{z}}=2\mu\,dh\otimes
g^{\mathcal{L}^{z}}\;.$ (2.14)

A general $\eta$ is locally exact, and therefore $\mathcal{L}^{z}$ can be
locally described as above. Thus $d_{z}=d^{\mathcal{L}^{z}}$ on
$\Omega(M)=\Omega(M,\mathcal{L}^{z})$, and (2.14) gives

$\nabla^{\mathcal{L}^{z}}g^{\mathcal{L}^{z}}=2\mu\eta\otimes
g^{\mathcal{L}^{z}}\;.$ (2.15)

####  2.1.5. Perturbed operators on oriented manifolds

Using for instance the interpretation of $d_{z}$ given in Section 2.1.4, the
mapping $(\alpha,\beta)\mapsto\alpha\wedge\beta$ induces a bilinear map
$H_{z}^{k}(M)\times H_{-z}^{l}(M)\to H^{k+l}(M)$, and the mapping
$(\alpha,\beta)\mapsto\alpha\wedge\bar{\beta}$ induces a sesquilinear map
$H_{z}^{k}(M)\times H_{-\bar{z}}^{l}(M)\to H^{k+l}(M)$.

In the rest of this subsection, assume $M$ is oriented. Therefore the above
maps and integration on $M$ define a nondegenerate bilinear pairing
$H_{z}^{k}(M)\times H_{-z}^{n-k}(M)\to\mathbb{C}$, and a nondegenerate
sesquilinear pairing $H_{z}^{k}(M)\times H_{-\bar{z}}^{n-k}(M)\to\mathbb{C}$.
Thus
$\beta_{z}^{k}=\beta_{-z}^{n-k}=\beta_{-\bar{z}}^{n-k}=\beta_{\bar{z}}^{k}$.

Let $\star$ denote the $\mathbb{C}$-linear extension to $\Lambda M$ of the
Hodge operator $\star$ on $\Lambda_{\mathbb{R}}M$, which is unitary, and let
$\bar{\star}$ denote its $\mathbb{C}$-anti-linear extension. These operators
are determined by the conditions

$\alpha\wedge\overline{\star\beta}=g(\alpha,\beta)\,\operatorname{vol}=\alpha\wedge\bar{\star}\beta$

for $\alpha,\beta\in\Omega(M)$, where $\operatorname{vol}=\star 1$ is the
volume form. The following equalities on $\Omega^{k}(M)$ follow from (2.3) and
the usual equalities relating $\star$, $d$, $\delta$, ${\eta\wedge}$ and
${\eta\lrcorner}$ (see e.g. [61, Chapters 1 and 3], [29, Section 1.5.2], [6,
Section 3.6]):

$\left.\begin{array}[]{c}d_{z}\,\star=(-1)^{k}\star\,\delta_{-\bar{z}}\;,\quad\delta_{z}\,\star=(-1)^{k+1}\star\,d_{-\bar{z}}\;,\quad\Delta_{z}\,\star=\star\,\Delta_{-\bar{z}}\;,\\\\[4.0pt]
d_{z}\,\bar{\star}=(-1)^{k}\;\bar{\star}\;\delta_{-z}\;,\quad\delta_{z}\;\bar{\star}=(-1)^{k+1}\,\bar{\star}\;d_{-z}\;,\quad\Delta_{z}\;\bar{\star}=\bar{\star}\;\Delta_{-z}\;.\end{array}\right\\}$
(2.16)

Then we get a linear isomorphism
$\star:\ker\Delta_{z}\to\ker\Delta_{-\bar{z}}$ and an anti-linear isomorphism
$\bar{\star}:\ker\Delta_{z}\to\ker\Delta_{-z}$, inducing a linear isomorphism
$H_{z}^{k}(M)\cong H_{-\bar{z}}^{n-k}(M)$ and an anti-linear isomorphism
$H_{z}^{k}(M)\cong H_{-z}^{n-k}(M)$ by (2.7).

###  2.2. Perturbation of the Sobolev norms

For $m\in\mathbb{N}_{0}$ and $\omega\in Z^{1}(M)$, define the norm $\|\
\|_{m,\omega}$ on $H^{m}(M;\Lambda)$ by

$\|\alpha\|_{m,\omega}=\sum_{k=0}^{m}\big{\|}D_{\omega}^{k}\alpha\big{\|}\;.$

######  Proposition 2.1.

For all $\omega\in Z^{1}(M)$ and $\alpha\in H^{m}(M;\Lambda)$,

$\displaystyle\|\alpha\|_{m,\omega}\leq
C_{m}\sum_{k=0}^{m}\|\omega\|_{C^{k}}^{m-k}\|\alpha\|_{k}\;,\quad\|\alpha\|_{m}\leq
C_{m}\sum_{k=0}^{m}\|\omega\|_{C^{k}}^{m-k}\|\alpha\|_{k,\omega}\;.$

###### Proof.

We proceed by induction on $m$. We have $\|\ \|_{0,\omega}=\|\ \|$. Now take
$m>0$ and assume these inequalities hold for $m-1$. For $\eta\in
Z^{1}(M,\mathbb{R})$ and $\alpha\in\Omega(M)$, we have

$\|\hat{c}(\eta)\alpha\|_{m},\|c(\eta)\alpha\|_{m}\leq
C^{\prime}_{m}\|\eta\|_{C^{m}}\|\alpha\|_{m}\;.$ (2.17)

Applying these inequalities to the real and imaginary parts of $\omega$, and
using the induction hypothesis and (2.3), we get

$\displaystyle\|\alpha\|_{m,\omega}$
$\displaystyle=\|\alpha\|+\|D_{\omega}\alpha\|_{m-1,\omega}\leq\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\|\omega\|_{C^{k}}^{m-1-k}\|D_{\omega}\alpha\|_{k}$
$\displaystyle\leq\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\|\omega\|_{C^{k}}^{m-1-k}\big{(}\|D\alpha\|_{k}+C^{\prime}_{k}\|\omega\|_{C^{k}}\|\alpha\|_{k}\big{)}$
$\displaystyle\leq\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\|\omega\|_{C^{k}}^{m-1-k}\big{(}\|\alpha\|_{k+1}+C^{\prime}_{k}\|\omega\|_{C^{k}}\|\alpha\|_{k}\big{)}$
$\displaystyle\leq C_{m}\sum_{l=0}^{m}\|\eta\|_{C^{l}}^{m-l}\|\alpha\|_{l}\;,$

and

$\displaystyle\|\alpha\|_{m}$
$\displaystyle=\|\alpha\|+\|D\alpha\|_{m-1}\leq\|\alpha\|+\|D_{\omega}\alpha\|_{m-1}+C^{\prime}_{m-1}\|\omega\|_{C^{m-1}}\|\alpha\|_{m-1}$
$\displaystyle\leq\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\big{(}\|\omega\|_{C^{k}}^{m-1-k}\|D_{\omega}\alpha\|_{k,\omega}+C^{\prime}_{m-1}\|\omega\|_{C^{k}}^{m-k}\|\alpha\|_{k,\omega}\big{)}$
$\displaystyle\leq\|\alpha\|+C_{m-1}\sum_{k=0}^{m-1}\big{(}\|\omega\|_{C^{k}}^{m-1-k}\|\alpha\|_{k+1,\omega}+C^{\prime}_{m-1}\|\eta\|_{C^{k}}^{m-k}\|\alpha\|_{k,\omega}\big{)}$
$\displaystyle\leq
C_{m}\sum_{l=0}^{m}\|\omega\|_{C^{l}}^{m-l}\|\alpha\|_{l,\omega}\;.\qed$

Let $Z(M,\mathbb{Z})\subset Z(M,\mathbb{R})$ denote the graded subspace of
forms that represent cohomology classes in the image of the canonical
homomorphism $H^{\bullet}(M,\mathbb{Z})\to H^{\bullet}(M,\mathbb{R})$. Recall
that we can consider $H^{1}(M,\mathbb{Z})$ as a lattice in
$H^{1}(M,\mathbb{R})$ by the universal coefficient theorem for cohomology. Let
$\theta$ be the multivalued angle function on $\mathbb{S}^{1}$. Then $d\theta$
is the angular form on $\mathbb{S}^{1}$ with
$\int_{\mathbb{S}^{1}}d\theta=2\pi$. For $\eta\in Z^{1}(M,\mathbb{R})$, we
have $\eta\in 2\pi Z^{1}(M,\mathbb{Z})$ if and only if there is some smooth
map $h:M\to\mathbb{S}^{1}$ such that $\eta=h^{*}d\theta$ (see e.g. [26, Lemma
2.1]).

In Proposition 2.1, the dependence of the constants on $\omega$ cannot be
avoided. For instance, for $M=\mathbb{S}^{1}$ with the standard metric
$g=(d\theta)^{2}$, we have $\|1\|_{m}=\sqrt{2\pi}$, whereas
$\|1\|_{m,i\eta}=\sqrt{2\pi}\sum_{k=0}^{m}|\nu|^{k}$ for $\eta=\nu\,d\theta$
($\nu\in\mathbb{R}$). However, the following version of a Sobolev inequality
for $\|\ \|_{m,i\eta}$ involves a constant independent of $\eta$.

######  Proposition 2.2.

If $m>n/2$, for all $\eta\in Z^{1}(M,\mathbb{R})$ and $\alpha\in
H^{m}(M;\Lambda)$,

$\|\alpha\|_{L^{\infty}}\leq C_{m}\|\alpha\|_{m,i\eta}\;.$

###### Proof.

By the Sobolev embedding theorem, we have

$C_{m,i\eta}:=\sup_{0\neq\alpha\in\Omega(M)}\frac{\|\alpha\|_{L^{\infty}}}{\|\alpha\|_{m,i\eta}}>0\;.$

Take any $\eta\in Z^{1}(M,\mathbb{R})$ and $\omega\in 2\pi
Z^{1}(M,\mathbb{Z})$, and let $\eta^{\prime}=\eta+\omega$. Then
$\omega=h^{*}d\theta$ for some smooth function $h:M\to\mathbb{S}^{1}$. Since
the difference between the multiple values of $\theta$ at every point of
$\mathbb{S}^{1}$ are in $2\pi\mathbb{Z}$, the functions $e^{\pm ih^{*}\theta}$
are well defined and smooth on $M$. Moreover, applying (2.12) locally, we get
$D_{i\eta^{\prime}}=e^{-ih^{*}\theta}\,D_{i\eta}\,e^{ih^{*}\theta}$. So, for
$0\neq\alpha\in\Omega(M)$,

$\displaystyle\|\alpha\|_{L^{\infty}}$
$\displaystyle=\|e^{i\,h^{*}\theta}\alpha\|_{L^{\infty}}\leq
C_{m,i\eta}\|e^{ih^{*}\theta}\alpha\|_{m,i\eta}$
$\displaystyle=C_{m,i\eta}\sum_{k=0}^{m}\|D_{i\eta}^{k}\,e^{ih^{*}\theta}\alpha\|=C_{m,i\eta}\sum_{k=0}^{m}\|e^{-ih^{*}\theta}\,D_{i\eta}^{k}\,e^{ih^{*}\theta}\alpha\|$
$\displaystyle=C_{m,i\eta}\sum_{k=0}^{m}\|D_{i\eta^{\prime}}^{k}\alpha\|=C_{m,i\eta}\|\alpha\|_{m,i\eta^{\prime}}\;.$

This shows that

$\sup_{\eta^{\prime}\in\eta+2\pi Z^{1}(M,\mathbb{Z})}C_{m,i\eta^{\prime}}\leq
C_{m,i\eta}\;.$ (2.18)

Since $2\pi H^{1}(M,\mathbb{Z})$ is a lattice in $H^{1}(M,\mathbb{R})$, there
is a compact subset $K\subset H^{1}(M,\mathbb{R})$ such that

$K+2\pi H^{1}(M,\mathbb{Z})=H^{1}(M,\mathbb{R})\;.$ (2.19)

Take a linear subspace $V\subset Z^{1}(M,\mathbb{R})$ such that the canonical
projection $V\to H^{1}(M,\mathbb{R})$ is an isomorphism, and let $L\subset V$
be the compact subset that corresponds to $K$. By (2.19),

$L+2\pi Z^{1}(M,\mathbb{Z})=Z^{1}(M,\mathbb{R})\;.$ (2.20)

Moreover $L$ is bounded with respect to $\|\ \|_{C^{m}}$. Therefore, by
Proposition 2.1, for all $\eta\in L$ and $\alpha\in\Omega(M)$,

$\|\alpha\|_{L^{\infty}}\leq C_{m,0}\|\alpha\|_{m}\leq
C_{m}\|\alpha\|_{m,i\eta}\;,$

yielding

$\sup_{\eta\in L}C_{m,i\eta}\leq C_{m}\;.$ (2.21)

The result follows from (2.18), (2.20) and (2.21). ∎

Given $\eta\in Z^{1}(M,\mathbb{R})$, we write $\|\ \|_{m,z}=\|\ \|_{m,z\eta}$.
Proposition 2.1 has the following direct consequence.

######  Corollary 2.3.

For all $\alpha\in H^{m}(M;\Lambda)$,

$\|\alpha\|_{m,z}\leq
C_{m}\sum_{k=0}^{m}|z|^{m-k}\|\alpha\|_{k}\;,\quad\|\alpha\|_{m}\leq
C_{m}\sum_{k=0}^{m}|z|^{m-k}\|\alpha\|_{k,z}\;.$

######  Proposition 2.4.

For all $\alpha\in H^{1}(M;\Lambda)$,

$\|\alpha\|_{1,z}\leq
C\big{(}\|\alpha\|_{1,i\nu}+|\mu|\|\alpha\|\big{)}\;,\quad\|\alpha\|_{1,i\nu}\leq
C\big{(}\|\alpha\|_{1,z}+|\mu|\|\alpha\|\big{)}\;.$

###### Proof.

By (2.3) and (2.17),

$\displaystyle\|\alpha\|_{1,z}$
$\displaystyle=\|\alpha\|+\|D_{z}\alpha\|\leq\|\alpha\|+\|D_{i\nu}\alpha\|+C^{\prime}|\mu|\|\alpha\|$
$\displaystyle\leq C\big{(}\|\alpha\|_{1,i\nu}+|\mu|\|\alpha\|\big{)}\;,$
$\displaystyle\|\alpha\|_{1,i\nu}$
$\displaystyle=\|\alpha\|+\|D_{i\nu}\alpha\|\leq\|\alpha\|+\|D_{z}\alpha\|+C^{\prime}|\mu|\|\alpha\|$
$\displaystyle\leq C\big{(}\|\alpha\|_{1,z}+|\mu|\|\alpha\|\big{)}\;.\qed$

##  3\. Small and large complexes

###  3.1. Preliminaries on Morse forms

Recall that a critical point $p$ of any $h\in C^{\infty}(M,\mathbb{R})$ is
called _nondegenerate_ if $\operatorname{Hess}_{p}h$ is a non-degenerate
symmetric bilinear form on $T_{p}M$. The index of $\operatorname{Hess}_{p}h$,
denoted by $\operatorname{ind}(p)$, is called the _index_ of $h$ at $p$. By
the Morse lemma [46, Lemma 2.2], this means that

$h-h(p)=\frac{1}{2}\sum_{j=1}^{n}\epsilon_{p,j}(x_{p}^{j})^{2}$ (3.1)

on the domain $U_{p}$ of some coordinates $x_{p}=(x_{p}^{1},\dots,x_{p}^{n})$
(centered) at $p$, called _Morse coordinates_ , where

$\epsilon_{p,j}=\begin{cases}-1&\text{if $j\leq\operatorname{ind}(p)$}\\\
1&\text{if $j>\operatorname{ind}(p)$}\;.\end{cases}$ (3.2)

Recall that $h$ is called a _Morse function_ when all of its critical points
are nondegenerate. In this case, its critical points form a finite set denoted
by $\operatorname{Crit}(h)$. The Morse functions form an open and dense subset
of $C^{\infty}(M,\mathbb{R})$ [34, Theorem 6.1.2]. On every $U_{p}$, we can
assume the metric is Euclidean with respect to Morse coordinates:

$g=\sum_{j=1}^{n}(dx_{p}^{j})^{2}\;.$ (3.3)

Now take any $\eta\in Z^{1}(M,\mathbb{R})$. A zero $p$ of $\eta$ is said to be
_nondegenerate_ if $\eta=dh_{\eta,p}$ around $p$ for some local Morse function
$h_{\eta,p}$, which is chosen so that $h_{\eta,p}(p)=0$. The index
$\operatorname{ind}(p)$ of $h_{\eta,p}$ at $p$ is also called the _index_ of
$\eta$ at $p$. On the domain $U_{p}$ of Morse coordinates
$x_{p}=(x_{p}^{1},\dots,x_{p}^{n})$ for $h_{\eta,p}$ at $p$, $h_{\eta,p}$ is
given by the right-hand sides of (3.1), and

$\eta=\sum_{j=1}^{n}\epsilon_{p,j}x_{p}^{j}\,dx_{p}^{j}\;.$ (3.4)

It is also said that $x_{p}=(x_{p}^{1},\dots,x_{p}^{n})$ are _Morse
coordinates_ for $\eta$ at $p$. With the notation

$x_{p}^{-}=(x_{p}^{1},\dots,x_{p}^{k}):U_{p}\to\mathbb{R}^{k}\;,\quad
x_{p}^{+}=(x_{p}^{k+1},\dots,x_{p}^{n}):U_{p}\to\mathbb{R}^{n-k}\;,$

for $k=\operatorname{ind}(p)$, we can write

$h_{\eta,p}=\frac{1}{2}\big{(}|x_{p}^{+}|^{2}-|x_{p}^{-}|^{2}\big{)}\;.$ (3.5)

If all zeros are nondegenerate, then $\eta$ is called a _Morse form_. In this
case, its zeros form a finite set, $\mathcal{X}=\operatorname{Zero}(\eta)$;
subsets of $\mathcal{X}$ defined by conditions on the index are denoted by
writing the conditions as subscripts; for instance, $\mathcal{X}_{k}$,
$\mathcal{X}_{+}$ and $\mathcal{X}_{<k}$ are the subsets of zeros of index
$k$, of positive index, and of index${}<k$, respectively. For any $\xi\in
H^{1}(M,\mathbb{R})$, the Morse representatives of $\xi$ form a dense open
subset of $\xi$, considered as a subspace of $\Omega^{1}(M,\mathbb{R})$ with
the $C^{\infty}$ topology (see e.g. [57, Theorem 2.1.25]).

From now on, unless otherwise stated, we will use some $\eta\in
Z^{1}(M,\mathbb{R})$ and a Riemannian metric $g$ on $M$ such that $\eta$ is a
Morse form, and $g$ is Euclidean with respect to Morse coordinates around zero
points of $\eta$; i.e., $\eta$ and $g$ satisfy a (Section 1.1).

The Hopf index of $-\eta^{\sharp}$ at every $p\in\mathcal{X}_{k}$ is
$(-1)^{k}$ (see Section 5.1.1). Thus, by the Hopf index theorem for
$-\eta^{\sharp}$,

$\sum_{k=0}^{n}(-1)^{k}|\mathcal{X}_{k}|=\chi(M)\;.$ (3.6)

###  3.2. The small and large spectrum

Consider the perturbed operators (2.3) defined by $\eta$ and $g$. Recall that
$\mathcal{X}$ denotes the zero set of $\eta$. We can suppose the closures of
the domains of Morse coordinates, $\overline{U_{p}}$ ($p\in\mathcal{X}$), are
disjoint from each other, and $x_{p}(U_{p})=(-4r,4r)^{n}$ for some $r>0$
independent of $p$. Let $U=\bigcup_{p\in\mathcal{X}}U_{p}$.

Denoting also the coordinates of $\mathbb{R}^{n}$ by
$(x_{p}^{1},\dots,x_{p}^{n})$, consider the function $h_{p}\in
C^{\infty}(\mathbb{R}^{n})$ defined by the right-hand side of (3.1) (or
(3.5)). Let $d^{\prime}_{p,z}$, $\delta^{\prime}_{p,z}$, $D^{\prime}_{p,z}$
and $\Delta^{\prime}_{p,z}$ ($z\in\mathbb{C}$) denote the corresponding
Witten’s operators on $\mathbb{R}^{n}$, whose restrictions to $(-4r,4r)^{n}$
agree via $x_{p}$ with the restrictions of $d_{\mu}$, $\delta_{\mu}$,
$D_{\mu}$ and $\Delta_{\mu}$ to $U_{p}$.

######  Proposition 3.1 (See e.g. [61, Chapters 9 and 14], [71, Sections 4.5
and 4.7]).

The following holds for $\mu\in\mathbb{R}$:

  1. (i)

We have

$\Delta^{\prime}_{p,\mu}=\sum_{j=1}^{n}\Big{(}-\Big{(}\frac{\partial}{\partial
x_{p}^{j}}\Big{)}^{2}+\mu^{2}(x_{p}^{j})^{2}+\mu\epsilon_{p,j}[{dx_{p}^{j}\lrcorner},{dx_{p}^{j}\wedge}]\Big{)}\;.$
(3.7)

Moreover, using multi-index notation,

$[{dx_{p}^{j}\lrcorner},{dx_{p}^{j}\wedge}]dx_{p}^{J}=\begin{cases}dx_{p}^{J}&\text{if
$j\in J$}\\\ -dx_{p}^{J}&\text{if $j\notin J$}\;.\end{cases}$

  2. (ii)

$\Delta^{\prime}_{p,\mu}$ is a non-negative selfadjoint operator in
$L^{2}(\mathbb{R}^{n};\Lambda)$ with a discrete spectrum, which consists of
the eigenvalues

$\mu\sum_{j=1}^{n}(1+2u_{j}+\epsilon_{p,j}v_{j})\;,$ (3.8)

where $u_{j}\in\mathbb{N}_{0}$ and $v_{j}=\pm 1$. For the restriction of
$\Delta^{\prime}_{p,\mu}$ to $k$-forms, the spectrum has the additional
requirement that exactly $k$ of the numbers $v_{j}$ are equal to $1$. In
particular, $0$ is an eigenvalue of $\Delta^{\prime}_{p,\mu}$ with
multiplicity $1$ (choosing $u_{j}=0$ and $v_{j}=-\epsilon_{p,j}$ for all $j$),
and the nonzero eigenvalues are of order $\mu$ as $\mu\to+\infty$.
$D^{\prime}_{p,\mu}$ is also a selfadjoint operator in
$L^{2}(\mathbb{R}^{n};\Lambda)$ with a discrete spectrum, which consists of
the positive and negative square roots of (3.8).

  3. (iii)

The kernel of $D^{\prime}_{p,\mu}$ and $\Delta^{\prime}_{p,\mu}$ is generated
by the normalized form

$e^{\prime}_{p,\mu}=\Big{(}\frac{\mu}{\pi}\Big{)}^{n/4}e^{-\mu|x_{p}|^{2}/2}\,dx_{p}^{1}\wedge\dots\wedge
dx_{p}^{\operatorname{ind}(p)}\;.$

For any $z\in\mathbb{C}$ with $\mu>0$, let $\Delta^{\prime}_{p,z}=e^{-i\nu
h_{p}}\Delta^{\prime}_{p,\mu}e^{i\nu h_{p}}$. Since the operator of
multiplication by $e^{-i\nu h_{p}}$ is unitary, $\Delta^{\prime}_{p,z}$ is
also selfadjoint and non-negative in $L^{2}(\mathbb{R}^{n};\Lambda)$, it has a
discrete spectrum with the same eigenvalues and multiplicities as
$\Delta^{\prime}_{p,\mu}$, and its kernel is generated by the normalized form
$e^{\prime}_{p,z}:=e^{-i\nu h_{p}}e^{\prime}_{p,\mu}$. We will also use the
notation

$e^{\prime}_{p,z}=x_{p}^{*}e^{\prime}_{p,z}\in
C^{\infty}\big{(}U_{p};\Lambda^{\operatorname{ind}(p)}\big{)}\;.$

The function $x_{p}^{*}h_{p}\in C^{\infty}(U_{p})$ agrees with $h_{\eta,p}$,
which is also denoted by $h_{p}$ in this section.

Fix an even $C^{\infty}$ function $\rho:\mathbb{R}\to[0,1]$ such that $\rho=1$
on $[-r,r]$ and $\operatorname{supp}\rho\subset[-2r,2r]$. For every
$p\in\mathcal{X}$, let

$\displaystyle\rho_{p}$ $\displaystyle=\rho(x_{p}^{1})\cdots\rho(x_{p}^{n})\in
C^{\infty}_{\text{\rm c}}(U_{p})\;,$ (3.9) $\displaystyle e_{p,\mu}$
$\displaystyle=\frac{\rho_{p}}{a_{\mu}}e^{\prime}_{p,\mu}\in
C^{\infty}_{\text{\rm
c}}\big{(}U_{p};\Lambda^{\operatorname{ind}(p)}\big{)}\;,$ (3.10)
$\displaystyle e_{p,z}$ $\displaystyle=e^{-i\nu
h_{p}}e_{p,\mu}=\frac{\rho_{p}}{a_{\mu}}e^{\prime}_{p,z}\in
C^{\infty}_{\text{\rm
c}}\big{(}U_{p};\Lambda^{\operatorname{ind}(p)}\big{)}\;,$ (3.11)

where

$a_{\mu}=\bigg{(}\int_{-2r}^{2r}\rho(x)^{2}e^{-\mu
x^{2}}\,dx\bigg{)}^{n/2}=\Big{(}\frac{\pi}{\mu}\Big{)}^{n/4}+O(e^{-c\mu})\;,$
(3.12)

as $\mu\to+\infty$. The extensions by zero of the forms $e_{p,z}$ to $M$ are
also denoted by $e_{p,z}$. They form an orthonormal basis of a graded subspace
$E_{z}\subset\Omega(M)$ with $\dim E_{z}=|\mathcal{X}|$. Let $P_{z}$ be the
orthogonal projection of $L^{2}(M;\Lambda)$ to $E_{z}$; thus $P_{z}^{\perp}$
is the orthogonal projection to $E_{z}^{\perp}$.

######  Remark 3.2.

For the sake of simplicity, most of our results are stated for $\mu\gg 0$ or
as $\mu\to+\infty$, but they have obvious versions for $\mu\ll 0$ or as
$\mu\to-\infty$, as follows by considering $-\eta$.

######  Proposition 3.3.

If $\mu\gg 0$ and $\beta\in H^{1}(M;\Lambda)$ with
$\operatorname{supp}\beta\subset M\setminus U$, then

$\|D_{z}\beta\|\geq C\mu\,\|\beta\|\;.$

###### Proof.

This follows like [71, Proposition 4.7], using that $\mathsf{H}_{\eta}$ is of
order zero in (2.3). Actually, according to the statement of [71, Proposition
4.7], this inequality would hold with $\sqrt{\mu}$ instead of $\mu$, but its
proof clearly shows that using $\mu$ is fine. ∎

######  Proposition 3.4.

The following properties hold:

  1. (i)

$P_{z}D_{z}P_{z}=0$.

  2. (ii)

If $\mu\gg 0$, $\alpha\in E_{z}$ and $\beta\in E_{z}^{\perp}\cap
H^{1}(M;\Lambda)$, then

$\|P_{z}^{\perp}D_{z}\alpha\|\leq
e^{-c\mu}\|\alpha\|\;,\quad\|P_{z}D_{z}\beta\|\leq e^{-c\mu}\|\beta\|\;.$

  3. (iii)

If $\mu\gg 0$ and $\beta\in E_{z}^{\perp}\cap H^{1}(M;\Lambda)$, then

$\|P_{z}^{\perp}D_{z}\beta\|\geq C\sqrt{\mu}\,\|\beta\|\;.$

###### Proof.

This follows like [71, Propositions 4.11, 4.12 and 5.6]. Property i is true
because every $D_{z}e_{p,z}$ is supported in $U_{p}$ and has homogeneous
components of degree different from $\operatorname{ind}(p)$; therefore it is
orthogonal to $\ker\Delta_{z}$. The other properties are consequences of
Propositions 3.1 and 3.3 and (3.9)–(3.12). According to [71, Proposition
4.11], the inequalities of ii hold with $1/\mu$ instead of $e^{-c\mu}$, but
its proof shows that indeed $e^{-c\mu}$ can be achieved. ∎

######  Proposition 3.5.

For all $m\in\mathbb{N}_{0}$, if $\mu\gg 0$, then

$\|D_{z}e_{p,z}\|_{m},\|D_{z}e_{p,z}\|_{m,i\nu}\leq e^{-c_{m}\mu}\;.$

###### Proof.

From Proposition 3.1 iii, (2.1), (3.10) and (3.11), we get

$D_{z}e_{p,z}=D_{z}\Big{(}\frac{\rho_{p}}{a_{\mu}}e^{\prime}_{p,z}\Big{)}=e^{-i\nu
h_{p}}\frac{1}{a_{\mu}}\Big{(}\frac{\pi}{4}\Big{)}^{n/4}\hat{c}(d\rho_{p})e^{\prime}_{p,\mu}\;.$
(3.13)

Thus the stated estimate of $\|D_{z}e_{p,z}\|_{m}$ is true by (3.10) and
(3.12), and since $d\rho_{p}=0$ around $p$. (When $\nu=0$, this is indicated
in [71, Eq. (6.17)].)

By (2.12), for all $k\in\mathbb{N}_{0}$ and $p\in\mathcal{X}$, the form
$D_{i\nu}^{k}D_{z}e_{p,z}$ is the extension by zero of the form $e^{-i\nu
h_{p}}D^{k}D_{\mu}e_{p,\mu}$ on $U_{p}$. Then the stated estimate of
$\|D_{z}e_{p,z}\|_{m,i\nu}$ follows from the case $\nu=0$. ∎

######  Corollary 3.6.

If $\mu\gg 0$, then

$\|D_{z}e_{p,z}\|_{L^{\infty}}\leq e^{-c\mu}\;.$

###### Proof.

Apply Propositions 2.2 and 3.5. ∎

Consider the partition of $\operatorname{spec}\Delta_{z}$ into its
intersections with $[0,1]$ and $(1,\infty)$, called the _small_ and _large
spectrum_ ; the term _small/large eigenvalues_ may be also used. Let
$E_{z,\text{\rm sm}}\subset\Omega(M)$ denote the graded finite dimensional
subspace generated by the eigenforms of the small eigenvalues, let
$E_{z,\text{\rm la}}=E_{z,\text{\rm sm}}^{\perp}$ in $L^{2}(M;\Lambda)$, and
let $P_{z,\text{\rm sm/la}}$ be the orthogonal projections to $E_{z,\text{\rm
sm/la}}$. Note that $E_{z,\text{\rm sm}}$ and $E_{z,\text{\rm
la}}\cap\Omega(M)$ are subcomplexes with $d_{z}$, called the _small_ and
_large complexes_ , and the term _small/large projection_ will be used for
$P_{z,\text{\rm sm/la}}$. Thus $(\Omega(M),d_{z})$ splits into a topological
direct sum of the subcomplexes $E_{z,\text{\rm sm}}$ and $E_{z,\text{\rm
la}}\cap\Omega(M)$, and (2.6) gives

$H^{\bullet}(E_{z,\text{\rm sm}},d_{z})\equiv H_{z}^{\bullet}(M)\;,\quad
H^{\bullet}(E_{z,\text{\rm la}}\cap\Omega(M),d_{z})=0\;.$ (3.14)

For any operator $B$ defined on $\Omega(M)$ or $L^{2}(M;\Lambda)$, let
$B_{z,\text{\rm sm/la}}=BP_{z,\text{\rm sm/la}}$.

######  Proposition 3.7.

For all $m\in\mathbb{N}_{0}$, $\mu\gg 0$ and $\alpha\in E_{z}$,

$\|\alpha-P_{z,\text{\rm sm}}\alpha\|_{m,i\nu}\leq e^{-c_{m}\mu}\|\alpha\|\;.$

###### Proof.

This follows like [71, Lemma 5.8 and Theorem 6.7], using $\|\ \|_{m,i\nu}$
instead of $\|\ \|_{m}$. The following are the main steps of the proof.

Let $\mathbb{S}^{1}=\\{\,\omega\in\mathbb{C}\mid|\omega|=1\,\\}$. With the
argument of the proof of [71, Eq. (5.27)], using Proposition 3.4, we get that,
for all $\alpha\in H^{1}(M;\Lambda)$, $w\in\mathbb{S}^{1}$ and $\mu\gg 0$,

$\|(w-D_{z})\alpha\|\geq C\|\alpha\|\;.$

Thus $w-D_{z}:H^{1}(M;\Lambda)\to L^{2}(M;\Lambda)$ has a bounded inverse and,
for all $\beta\in L^{2}(M;\Lambda)$, $w\in\mathbb{S}^{1}$ and $\mu\gg 0$,

$\big{\|}(w-D_{z})^{-1}\beta\big{\|}\leq C^{-1}\|\beta\|\;.$ (3.15)

On the other hand, arguing like in the proof of [71, Eq. (6.18)], it follows
that, for all $\gamma\in H^{m}(M;\Lambda)$, $w\in\mathbb{S}^{1}$ and $\mu\gg
0$,

$\|\gamma\|_{m,i\nu}\leq
C_{m}\big{(}\|(w-D_{z})\gamma\big{\|}_{m-1,i\nu}+\mu\|\gamma\|_{m-1,i\nu}+\|\gamma\|\big{)}\;.$

Continuing by induction on $m\in\mathbb{N}_{0}$, we obtain

$\|\gamma\|_{m,i\nu}\leq
C_{m}\Big{(}\mu^{m}\|\gamma\|+\sum_{k=1}^{m}\mu^{k-1}\|(w-D_{z})\gamma\big{\|}_{m-k,i\nu}\Big{)}\;.$

In other words, for all $\beta\in H^{m-1}(M;\Lambda)$,

$\big{\|}(w-D_{z})^{-1}\beta\big{\|}_{m,i\nu}\leq
C_{m}\Big{(}\mu^{m}\big{\|}(w-D_{z})^{-1}\beta\big{\|}+\sum_{k=1}^{m}\mu^{k-1}\|\beta\|_{m-k,i\nu}\Big{)}\;.$

Applying (3.15) to this inequality, we get

$\big{\|}(w-D_{z})^{-1}\beta\big{\|}_{m,i\nu}\leq
C_{m}\mu^{m}\|\beta\|_{m-1,i\nu}\;.$ (3.16)

Then, by Proposition 3.5,

$\big{\|}(w-D_{z})^{-1}D_{z}e_{p,z}\big{\|}_{m,i\nu}=O\big{(}e^{-c_{m}\mu}\big{)}$
(3.17)

as $\mu\to+\infty$, uniformly on $w\in\mathbb{S}^{1}$. But, endowing
$\mathbb{S}^{1}$ with the counter-clockwise orientation, basic spectral theory
gives (see e.g. [23, Section VII.3])

$\displaystyle P_{z,\text{\rm sm}}e_{p,z}-e_{p,z}$
$\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{S}^{1}}\big{(}(w-D_{z})^{-1}-w^{-1}\big{)}e_{p,z}\,dw$
$\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{S}^{1}}w^{-1}(w-D_{z})^{-1}D_{z}e_{p,z}\,dw\;.$ (3.18)

The result follows using (3.17) in (3.18). ∎

######  Corollary 3.8.

For $\mu\gg 0$ and $\alpha\in E_{z}$,

$\|\alpha-P_{z,\text{\rm sm}}\alpha\|_{L^{\infty}}\leq e^{-c\mu}\|\alpha\|\;.$

###### Proof.

Apply Propositions 2.2 and 3.7.

Alternatively, the proof of Proposition 3.7 can be modified as follows to get
this result (some step of this alternative argument will be used later).
Iterating (3.16), we get

$\big{\|}(w-D_{z})^{-1}\beta\big{\|}_{m,i\nu}\leq
C^{\prime}_{m}\mu^{(m+1)m/2}\|\beta\|\;,$

for all $\beta\in L^{2}(M;\Lambda)$. Then, by Proposition 2.2,

$\big{\|}(w-D_{z})^{-1}\beta\big{\|}_{L^{\infty}}\leq
C\mu^{(m+1)m/2}\|\beta\|\;.$ (3.19)

Thus, by Proposition 3.5,

$\big{\|}(w-D_{z})^{-1}D_{z}e_{p,z}\big{\|}_{L^{\infty}}=O\big{(}e^{-c_{m}\mu}\big{)}$

as $\mu\to+\infty$. Finally, apply this expression in (3.18). ∎

######  Corollary 3.9.

If $\mu\gg 0$, then $P_{z,\text{\rm sm}}:E_{z}\to E_{z,\text{\rm sm}}$ is an
isomorphism; in particular, $\dim E_{z,\text{\rm sm}}=|\mathcal{X}|$ and $\dim
E_{z,\text{\rm sm}}^{k}=|\mathcal{X}_{k}|$.

###### Proof.

The result follows from Proposition 3.7 for $m=0$ like [71, Proposition 5.5].
∎

When $\mu\gg 0$, (3.6) also follows from Corollary 3.9, (2.9) and (3.14).

######  Theorem 3.10 (Cf. [15, Theorem 3]).

We have

$\operatorname{spec}\Delta_{z}\subset\big{[}0,e^{-c|\mu|}\big{]}\cup\big{[}C|\mu|,\infty\big{)}\;.$

###### Proof.

We can assume $\mu\geq 0$ according to Remark 3.2. By Propositions 3.4, 3.7
and 2.4, for all $\alpha\in E_{z}$,

$\displaystyle\|D_{z}P_{z,\text{\rm sm}}\alpha\|$
$\displaystyle\leq\|D_{z}\alpha\|+\|D_{z}(\alpha-P_{z,\text{\rm
sm}}\alpha)\|\leq\|D_{z}\alpha\|+\|\alpha-P_{z,\text{\rm sm}}\alpha\|_{1,z}$
$\displaystyle\leq\|P_{z}^{\perp}D_{z}\alpha\|+C(\mu\|\alpha-P_{z,\text{\rm
sm}}\alpha\|+\|\alpha-P_{z,\text{\rm sm}}\alpha\|_{1,i\nu})$
$\displaystyle\leq\big{(}e^{-c\mu}+C\big{(}\mu
e^{-c_{0}\mu}+e^{-c_{1}\mu}\big{)}\big{)}\|\alpha\|\;.$

Hence, by Corollary 3.9, for all $\beta\in E_{z,\text{\rm sm}}$,

$0\leq\langle\Delta_{z}\beta,\beta\rangle=\|D_{z}\beta\|^{2}\leq
e^{-c\mu}\,\|\beta\|^{2}\;.$

This shows that

$\operatorname{spec}\Delta_{z}\cap[0,1]\subset\big{[}0,e^{-c\mu}\big{]}\;.$
(3.20)

Now let $\phi\in E_{z,\text{\rm la}}\cap H^{1}(M;\Lambda)$, and write
$\alpha=P_{z}\phi\in E_{z}$ and $\beta=P_{z}^{\perp}\phi\in E_{z}^{\perp}\cap
H^{1}(M;\Lambda)$. By Proposition 3.7,

$\|\alpha\|^{2}=\langle\alpha,\phi\rangle=\langle\alpha-P_{z,\text{\rm
sm}}\alpha,\phi\rangle\leq\|\alpha-P_{z,\text{\rm sm}}\alpha\|\|\phi\|\leq
e^{-c_{0}\mu}\|\alpha\|\|\phi\|\;,$

yielding

$\|\alpha\|\leq e^{-c_{0}\mu}\|\phi\|\;.$

So

$\|\beta\|=\|\phi-\alpha\|\geq\|\phi\|-\|\alpha\|\geq\big{(}1-e^{-c_{0}\mu}\big{)}\|\phi\|\;.$

Then, by Proposition 3.4,

$\displaystyle\|D_{z}\phi\|$
$\displaystyle\geq\|D_{z}\beta\|-\|D_{z}\alpha\|\geq\|P_{z}^{\perp}D_{z}\beta\|-e^{-c\mu}\|\alpha\|$
$\displaystyle\geq
C\sqrt{\mu}\,\|\beta\|-e^{-c\mu}\|\phi\|\geq\big{(}C\sqrt{\mu}\big{(}1-e^{-c_{0}\mu}\big{)}-e^{-c\mu}\big{)}\|\phi\|\;.$

Therefore, for all $\phi\in E_{z,\text{\rm la}}\cap H^{1}(M;\Lambda)$,

$\langle\Delta_{z}\phi,\phi\rangle=\|D_{z}\phi\|^{2}\geq C\mu\|\phi\|^{2}\;.$

This proves that

$\operatorname{spec}\Delta_{z}\cap(1,\infty)\subset[C\mu,\infty)\;.$ (3.21)

The inclusions (3.20) and (3.21) give the result for $\mu\gg 0$. But, in those
inclusions, we can take $c$ and $C$ so small that, if one of them is not true
for some $\mu\geq 0$, then $C\mu\leq e^{-c\mu}$. ∎

###  3.3. Ranks of some projections in the small complex

Recall that $(\Pi_{z}^{\perp})_{\text{\rm sm},k}$, $\Pi^{1}_{z,\text{\rm
sm},k}$ and $\Pi^{2}_{z,\text{\rm sm},k}$ denote the orthogonal projections to
the images of $\Delta_{z,\text{\rm sm},k}$, $d_{z,\text{\rm sm},k-1}$ and
$\delta_{z,\text{\rm sm},k+1}$, respectively. Let $m_{z,k}$, $m_{z,k}^{1}$ and
$m_{z,k}^{2}$ be the corresponding ranks (or traces) of these projections.
They satisfy

$m_{z,k}=m_{z,k}^{1}+m_{z,k}^{2}\;,\quad m_{z,0}^{1}=m_{z,n}^{2}=0\;,\quad
m_{z,k}^{2}=m_{z,k+1}^{1}\;,$ (3.22)

where the last equality is true because
$d_{z}:\operatorname{im}\delta_{z}\to\operatorname{im}d_{z}$ is an
isomorphism. For $\mu\gg 0$, we have
$m_{z,k},m_{z,k}^{j}\leq|\mathcal{X}_{k}|$ by Corollary 3.9 and (3.22).

######  Lemma 3.11.

The numbers $m_{z,k}^{j}$ are determined by the numbers $m_{z,k}$.

###### Proof.

This follows from (3.22) with an easy induction argument on $k$. ∎

######  Lemma 3.12.

We have $m_{z,k}=|\mathcal{X}_{k}|-\beta_{z}^{k}$.

###### Proof.

This is a consequence of (2.6), (3.14) and Corollary 3.9. ∎

######  Corollary 3.13.

$\operatorname{Tr}^{\text{\rm s}}((\Pi_{z}^{\perp})_{\text{\rm sm}})=0$.

###### Proof.

By (2.9), (3.6) and Lemma 3.12,

$\operatorname{Tr}^{\text{\rm s}}\big{(}(\Pi_{z}^{\perp})_{\text{\rm
sm}}\big{)}=\sum_{k}(-1)^{k}|\mathcal{X}_{k}|-\sum_{k}(-1)^{k}\beta_{z}^{k}=\chi(M)-\chi(M)=0\;.\qed$

######  Lemma 3.14.

If $M$ is oriented, then, for $k=0,\dots,n$,

$m_{z,k}=m_{-\bar{z},n-k}=m_{-z,n-k}\;,\quad
m_{z,k}^{1}=m_{-\bar{z},n-k}^{2}=m_{-z,n-k}^{2}\;.$

###### Proof.

This is true because, by (2.16),

$\displaystyle(\Pi_{z}^{\perp})_{\text{\rm sm},k}\,\star$
$\displaystyle=\star\,(\Pi_{-\bar{z}}^{\perp})_{\text{\rm sm},n-k}\;,$
$\displaystyle\quad\Pi^{1}_{z,\text{\rm sm},k}\,\star$
$\displaystyle=\star\,\Pi^{2}_{-\bar{z},\text{\rm sm},n-k}\;,$
$\displaystyle(\Pi_{z}^{\perp})_{\text{\rm sm},k}\,\bar{\star}$
$\displaystyle=\bar{\star}\,(\Pi_{-z}^{\perp})_{\text{\rm sm},n-k}\;,$
$\displaystyle\quad\Pi^{1}_{z,\text{\rm sm},k}\,\bar{\star}$
$\displaystyle=\bar{\star}\,\Pi^{2}_{-z,\text{\rm sm},n-k}\;.\qed$

######  Corollary 3.15.

For $|\mu|\gg 0$, $m_{z,k}$ and $m^{j}_{z,k}$ only depend on
$|\mathcal{X}_{k}|$ and the class $\xi=[\eta]\in H^{1}(M,\mathbb{R})$.

###### Proof.

Apply (2.10) and Lemmas 3.11 and 3.12. ∎

By Corollary 3.15, we write $m_{k}=m_{z,k}$ and $m_{k}^{j}=m_{z,k}^{j}$ for
$|\mu|\gg 0$.

######  Corollary 3.16.

$\operatorname{Tr}^{\text{\rm s}}(\Pi_{z,\text{\rm sm}}^{j})=0$ if $M$ is
oriented, $n$ is even and $\mu\gg 0$.

###### Proof.

By Corollaries 3.13, 3.15 and 3.14, and since $n$ is even,

$\operatorname{Tr}^{\text{\rm s}}\big{(}\Pi^{1}_{z,\text{\rm
sm}}\big{)}=-\operatorname{Tr}^{\text{\rm s}}\big{(}\Pi^{2}_{z,\text{\rm
sm}}\big{)}=-\operatorname{Tr}^{\text{\rm s}}\big{(}\Pi^{1}_{z,\text{\rm
sm}}\big{)}\;.\qed$

###  3.4. Asymptotic properties of the small projection

######  Notation 3.17.

Consider a function $f(x)>0$ ($x>0$). When referring to vectors in Banach
spaces, the order notation $O(f(|\mu|))$ ($\mu\to\pm\infty$) will be used for
a family of vectors $v=v(z)$ ($z\in\mathbb{C}$) with $\|v(z)\|=O(f(|\mu|))$.
This notation applies e.g. to bounded operators between Banach spaces. We may
also consider this notation when the Banach spaces depend on $z$.

######  Proposition 3.18.

For every $\tau\in\mathbb{R}$, on $L^{2}(M;\Lambda)$, as $\mu\to+\infty$,

$\displaystyle P_{z,\text{\rm sm}}$
$\displaystyle=P_{z}+O\big{(}e^{-c\mu}\big{)}=P_{z,\text{\rm
sm}}P_{z+\tau,\text{\rm sm}}P_{z,\text{\rm sm}}+O\big{(}\mu^{-2}\big{)}$
$\displaystyle=P_{z+\tau,\text{\rm sm}}+O\big{(}\mu^{-1}\big{)}\;.$

###### Proof.

By Corollary 3.9, for $\mu\gg 0$, the elements $P_{z,\text{\rm sm}}e_{p,z}$
($p\in\mathcal{X}$) form a base of $E_{z,\text{\rm sm}}$. Applying the Gram-
Schmidt process to this base, we get an orthonormal base $\tilde{e}_{p,z}$. By
Proposition 3.7,

$\tilde{e}_{p,z}=e_{p,z}+O\big{(}e^{-c\mu}\big{)}\;.$ (3.23)

This gives the first equality of the statement: for any $\alpha\in
L^{2}(M;\Lambda)$,

$\displaystyle P_{z}\alpha$
$\displaystyle=\sum_{p\in\mathcal{X}}\langle\alpha,e_{p,z}\rangle
e_{p,z}=\sum_{p\in\mathcal{X}}\langle\alpha,\tilde{e}_{p,z}\rangle\tilde{e}_{p,z}+O\big{(}e^{-c\mu}\big{)}\|\alpha\|$
$\displaystyle=P_{z,\text{\rm
sm}}\alpha+O\big{(}e^{-c\mu}\big{)}\|\alpha\|\;.$

Since the sets $U_{p}$ ($p\in\mathcal{X}$) are disjoint one another, for
$p\neq q$ in $\mathcal{X}$,

$\langle e_{p,z},e_{q,z+\tau}\rangle=0\;.$ (3.24)

On the other hand, by (3.9)–(3.12), we can also assume

$\displaystyle\langle e_{p,z},e_{p,z+\tau}\rangle$ $\displaystyle=\langle
e^{-i\nu h_{p}}e_{p,\mu},e^{-i\nu h_{p}}e_{p,\mu+\tau}\rangle=\langle
e_{p,\mu},e_{p,\mu+\tau}\rangle$
$\displaystyle=\frac{(\mu(\mu+\tau))^{n/4}}{\pi^{n/2}}\big{\langle}\rho_{p}e^{-\mu|x_{p}|^{2}/2},\rho_{p}e^{-(\mu+\tau)|x_{p}|^{2}/2}\big{\rangle}+O\big{(}e^{-c\mu}\big{)}$
$\displaystyle=\frac{(\mu(\mu+\tau))^{n/4}}{\pi^{n/2}}\int_{\mathbb{R}^{n}}e^{-(\mu+\tau/2)|x_{p}|^{2}}\,dx_{p}+O\big{(}e^{-c\mu}\big{)}$
$\displaystyle=\frac{(\mu(\mu+\tau))^{n/4}}{(\mu+\tau/2)^{n/2}}+O\big{(}e^{-c\mu}\big{)}=1+O\big{(}\mu^{-2}\big{)}\;,$
(3.25)

where $dx_{p}=dx_{p}^{1}\dots dx_{p}^{n}=\operatorname{vol}(x_{p})$. Combining
(3.23) for $z$ and $z+\tau$ with (3.24) and (3.25), we obtain

$\displaystyle P_{z+\tau,\text{\rm sm}}\tilde{e}_{p,z}$
$\displaystyle=\sum_{q\in\mathcal{X}}\langle\tilde{e}_{p,z},\tilde{e}_{q,z+\tau}\rangle\tilde{e}_{q,z+\tau}$
$\displaystyle=\sum_{q\in\mathcal{X}}\langle e_{p,z},e_{q,z+\tau}\rangle
e_{q,z+\tau}+O\big{(}e^{-c\mu}\big{)}$
$\displaystyle=e_{p,z+\tau}+O\big{(}\mu^{-2}\big{)}=\tilde{e}_{p,z+\tau}+O\big{(}\mu^{-2}\big{)}\;.$
(3.26)

Repeating (3.26) interchanging the roles of $z$ and $z+\tau$, we get

$P_{z,\text{\rm sm}}P_{z+\tau,\text{\rm sm}}\tilde{e}_{p,z}=P_{z,\text{\rm
sm}}\tilde{e}_{p,z+\tau}+O\big{(}\mu^{-2}\big{)}=\tilde{e}_{p,z}+O\big{(}\mu^{-2}\big{)}\;.$

This gives the second equality of the statement: for any $\alpha\in
L^{2}(M;\Lambda)$,

$\displaystyle P_{z,\text{\rm sm}}\alpha$
$\displaystyle=\sum_{p\in\mathcal{X}}\langle\alpha,\tilde{e}_{p,z}\rangle\tilde{e}_{p,z}$
$\displaystyle=P_{z,\text{\rm sm}}P_{z+\tau,\text{\rm
sm}}\sum_{p\in\mathcal{X}}\langle\alpha,\tilde{e}_{p,z}\rangle\tilde{e}_{p,z}+O\big{(}\mu^{-2}\big{)}\|\alpha\|$
$\displaystyle=P_{z,\text{\rm sm}}\alpha+O\big{(}\mu^{-2}\big{)}\|\alpha\|\;.$

By (3.26),

$\displaystyle\|\tilde{e}_{p,z}-\tilde{e}_{p,z+\tau}\big{\|}^{2}$
$\displaystyle=\|\tilde{e}_{p,z}\|^{2}-2\Re\langle\tilde{e}_{p,z},\tilde{e}_{p,z+\tau}\rangle+\|\tilde{e}_{p,z+\tau}\|^{2}$
$\displaystyle=2-2\Re\langle P_{z+\tau,\text{\rm
sm}}\tilde{e}_{p,z},\tilde{e}_{p,z+\tau}\rangle$
$\displaystyle=2-2\Re\langle\tilde{e}_{p,z+\tau},\tilde{e}_{p,z+\tau}\rangle+O\big{(}\mu^{-2}\big{)}=O\big{(}\mu^{-2}\big{)}\;,$

which means

$\tilde{e}_{p,z}=\tilde{e}_{p,z+\tau}+O\big{(}\mu^{-1}\big{)}\;.$ (3.27)

The last stated equality follows from (3.26) and (3.27): for any $\alpha\in
L^{2}(M;\Lambda)$,

$\displaystyle P_{z,\text{\rm sm}}\alpha$
$\displaystyle=\sum_{p\in\mathcal{X}}\langle\alpha,\tilde{e}_{p,z}\rangle\tilde{e}_{p,z}=\sum_{p\in\mathcal{X}}\langle\alpha,\tilde{e}_{p,z+\tau}\rangle\tilde{e}_{p,z+\tau}+O\big{(}\mu^{-1}\big{)}\alpha$
$\displaystyle=P_{z+\tau,\text{\rm
sm}}\alpha+O\big{(}\mu^{-1}\big{)}\alpha\;.\qed$

######  Corollary 3.19.

For every $\tau\in\mathbb{R}$, on $L^{2}(M;\Lambda)$,

$d_{z+\tau,\text{\rm sm}}-d_{z+\tau}P_{z,\text{\rm
sm}}=O\big{(}\mu^{-1}\big{)}\quad(\mu\to+\infty)\;.$

###### Proof.

Since $d_{z+\tau}=d_{z}+\tau\,{\eta\wedge}$, it follows from Theorem 3.10 that
$d_{z+\tau}$ is bounded on $E_{z,\text{\rm sm}}+E_{z+\tau,\text{\rm sm}}$,
uniformly on $\mu\gg 0$. Hence, by Proposition 3.18,

$d_{z+\tau,\text{\rm sm}}-d_{z+\tau}P_{z,\text{\rm
sm}}=d_{z+\tau}(P_{z+\tau,\text{\rm sm}}-P_{z,\text{\rm
sm}})=O\big{(}\mu^{-1}\big{)}\;.\qed$

######  Proposition 3.20.

On $L^{2}(M;\Lambda)$,

$P_{z,\text{\rm sm}}\,{\eta\wedge},{\eta\wedge}\,P_{z,\text{\rm
sm}}=O\big{(}\mu^{-1/2}\big{)}\quad(\mu\to+\infty)\;.$

###### Proof.

Let $\alpha\in L^{2}(M;\Lambda^{k})$ and $p\in\mathcal{X}_{k+1}$. Using multi-
index notation, write $\alpha=\sum_{|J|=k}\alpha_{J}\,dx_{p}^{J}$ on $U_{p}$
with $\alpha_{J}\in L^{2}(U_{p})$. Let
$J_{j}=\\{1,\dots,k+1\\}\setminus\\{j\\}$ ($j=1,\dots,k+1$). By Proposition
3.1 iii, (3.10) and (3.11),

$\langle\eta\wedge\alpha,e_{p,z}\rangle\\\ \begin{aligned}
&=-\frac{1}{a_{\mu}}\sum_{j=1}^{k+1}\big{\langle}\alpha_{J_{j}}x_{p}^{j}\,dx_{p}^{j}\wedge
dx_{p}^{J_{j}},e^{-i\nu
h_{p}}\rho_{p}e^{-\mu|x_{p}|^{2}/2}\,dx_{p}^{1}\wedge\dots\wedge
dx_{p}^{k+1}\big{\rangle}\\\
&=(-1)^{j}\frac{1}{a_{\mu}}\sum_{j=1}^{k+1}\big{\langle}\alpha_{J_{j}}x_{p}^{j},e^{-i\nu
h_{p}}\rho_{p}e^{-\mu|x_{p}|^{2}/2}\big{\rangle}\;.\end{aligned}$

Hence

$\displaystyle|\langle\eta\wedge\alpha,e_{p,z}\rangle|$
$\displaystyle\leq\frac{1}{a_{\mu}}\sum_{j=1}^{k+1}\big{|}\big{\langle}\alpha_{J_{j}},e^{-i\nu
h_{p}}\rho_{p}x_{p}^{j}e^{-\mu|x_{p}|^{2}/2}\big{\rangle}\big{|}$
$\displaystyle\leq\|\alpha\|\frac{1}{a_{\mu}}\sum_{j=1}^{k+1}\big{\|}\rho_{p}x_{p}^{j}e^{-\mu|x_{p}|^{2}/2}\big{\|}\;,$
(3.28)

and

$\displaystyle\big{\|}\rho_{p}x_{p}^{j}e^{-\mu|x_{p}|^{2}/2}\big{\|}$
$\displaystyle=\Big{(}\int_{U_{p}}(\rho_{p}(x_{p})x_{p}^{j})^{2}e^{-\mu|x_{p}|^{2}}\,\operatorname{vol}(x_{p})\Big{)}^{1/2}$
$\displaystyle=\Big{(}\int_{-2r}^{2r}\rho(x)^{2}x^{2}e^{-\mu
x^{2}}\,dx\Big{)}^{\frac{1}{2}}\Big{(}\int_{-2r}^{2r}\rho(y)^{2}e^{-\mu
y^{2}}\,dy\Big{)}^{\frac{n-1}{2}}$
$\displaystyle=\Big{(}\int_{-2r}^{2r}x^{2}e^{-\mu
x^{2}}\,dx\Big{)}^{\frac{1}{2}}\Big{(}\int_{-2r}^{2r}e^{-\mu
y^{2}}\,dy\Big{)}^{\frac{n-1}{2}}+O\big{(}e^{-c\mu}\big{)}$
$\displaystyle=\frac{1}{(2\mu)^{1/2}}\Big{(}\frac{\pi}{\mu}\Big{)}^{n/4}+O\big{(}e^{-c\mu}\big{)}\;.$
(3.29)

By (3.12), (3.28) and (3.29),

$|\langle\eta\wedge\alpha,e_{p,z}\rangle|=\|\alpha\|\,O\big{(}\mu^{-1/2}\big{)}\;.$

Since

$P_{z}(\eta\wedge\alpha)=\sum_{p\in\mathcal{X}_{k+1}}\langle\eta\wedge\alpha,e_{p,z}\rangle
e_{p,z}\;,$

it follows that

$\|P_{z}(\eta\wedge\alpha)\|=\|\alpha\|\,O\big{(}\mu^{-1/2}\big{)}\;.$

This shows that $P_{z}\,{\eta\wedge}=O\big{(}\mu^{-1/2}\big{)}$, yielding
$P_{z,\text{\rm sm}}\,{\eta\wedge}=O\big{(}\mu^{-1/2}\big{)}$ by Proposition
3.18.

On the other hand, for $q\in\mathcal{X}_{k}$,

$\eta\wedge
e_{q,z}=\frac{\rho_{q}}{a_{\mu}}e^{-\mu|x_{p}|^{2}/2}\sum_{j=k+1}^{n}x_{q}^{j}\,dx_{q}^{j}\wedge
dx_{q}^{1}\wedge\dots\wedge dx_{q}^{k}\;.$

So

$\|\eta\wedge
e_{q,z}\|=\frac{1}{a_{\mu}}\Bigg{(}\sum_{j=k+1}^{n}\int_{U_{q}}(\rho_{q}(x_{q})x_{q}^{j})^{2}e^{-\mu|x_{p}|^{2}}\,\operatorname{vol}(x_{q})\Bigg{)}^{1/2}\;,$

which becomes $O\big{(}\mu^{-1/2}\big{)}$ like in the previous part of the
proof. Since

${\eta\wedge}\,P_{z}\alpha=\sum_{q\in\mathcal{X}_{k}}\langle\alpha,e_{q,z}\rangle\,\eta\wedge
e_{q,z}\;,$

we get ${\eta\wedge}\,P_{z}=O\big{(}\mu^{-1/2}\big{)}$, yielding
${\eta\wedge}\,P_{z,\text{\rm sm}}=O\big{(}\mu^{-1/2}\big{)}$ as before. ∎

###  3.5. Derivatives of the small projection

######  Remark 3.21.

For reasons of brevity, most of the results about derivatives are stated for
$\partial_{z}$, which may be simply denoted with a dot. But there are obvious
versions of those results for $\partial_{\bar{z}}$ with analogous proofs.

######  Proposition 3.22.

We have

$\operatorname{rank}\partial_{z}P_{z,\text{\rm sm}}\leq
2|\mathcal{X}|\quad(\mu\gg 0)\;,\quad\partial_{z}P_{z,\text{\rm
sm}}=O\big{(}\mu^{-1/2}\big{)}\quad(\mu\to+\infty)\;.$

###### Proof.

By (2.4) and Theorem 3.10, for $\mu\gg 0$ and every $\omega\in\mathbb{S}^{1}$,
a standard computation gives

$\partial_{z}\big{(}(w-D_{z})^{-1}\big{)}=(w-D_{z})^{-1}\,{\eta\wedge}\,(w-D_{z})^{-1}\;.$
(3.30)

Then, by (3.15), $\partial_{z}\big{(}(w-D_{z})^{-1}\big{)}$ defines an
operator on $L^{2}(M;\Lambda)$, bounded uniformly on $w\in\mathbb{S}^{1}$ and
$z\in\mathbb{C}$. By (3.15) and Proposition 3.20, we also get

$P_{z,\text{\rm
la/sm}}\partial_{z}\big{(}(w-D_{z})^{-1}\big{)}\,P_{z,\text{\rm sm/la}}\\\
=(w-D_{z})^{-1}P_{z,\text{\rm la/sm}}\,{\eta\wedge}\,P_{z,\text{\rm
sm/la}}(w-D_{z})^{-1}=O\big{(}\mu^{-1/2}\big{)}\;,$

uniformly on $w\in\mathbb{S}^{1}$.

On the other hand, applying again basic spectral theory, we obtain

$P_{z,\text{\rm sm}}=\frac{1}{2\pi i}\int_{\mathbb{S}^{1}}(w-D_{z})^{-1}\,dw$

for $\mu\gg 0$, yielding

$\dot{P}_{z,\text{\rm sm}}=\frac{1}{2\pi
i}\int_{\mathbb{S}^{1}}\partial_{z}\big{(}(w-D_{z})^{-1}\big{)}\,dw\;,$ (3.31)

which defines an operator on $L^{2}(M;\Lambda)$, bounded uniformly on $z$.

Using that $P_{z,\text{\rm sm}}$ is an orthogonal projection, the argument of
the proof of [6, Proposition 9.37] shows that

$\dot{P}_{z,\text{\rm sm}}=P_{z,\text{\rm la}}\dot{P}_{z,\text{\rm
sm}}P_{z,\text{\rm sm}}+P_{z,\text{\rm sm}}\dot{P}_{z,\text{\rm
sm}}P_{z,\text{\rm la}}\;.$ (3.32)

So $\operatorname{rank}\dot{P}_{z,\text{\rm sm}}\leq
2\operatorname{rank}P_{z,\text{\rm sm}}\leq 2|\mathcal{X}|$ by Corollary 3.9,
and

$\displaystyle\dot{P}_{z,\text{\rm sm}}$ $\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{S}^{1}}P_{z,\text{\rm
la}}\partial_{z}\big{(}(w-D_{z})^{-1}\big{)}\,P_{z,\text{\rm sm}}\,dw$
$\displaystyle\phantom{=\text{}}\text{}+\frac{1}{2\pi
i}\int_{\mathbb{S}^{1}}P_{z,\text{\rm
sm}}\partial_{z}\big{(}(w-D_{z})^{-1}\big{)}\,P_{z,\text{\rm
la}}\,dw=O\big{(}\mu^{-1/2}\big{)}\;.\qed$

######  Lemma 3.23.

For all $p\in\mathcal{X}$,

$\partial_{z}e_{p,z}=\bigg{(}\frac{n}{8\mu}-\frac{|x^{+}_{p}|^{2}}{2}+O(e^{-c\mu})\bigg{)}e_{p,z}\quad(\mu\to+\infty)\;.$

###### Proof.

Using integration by parts, and since $\rho$ is an even function and
$\rho^{\prime}$ vanishes on $[-r,r]$, we obtain

$\displaystyle\int_{-2r}^{2r}\rho(x)^{2}x^{2}e^{-\mu x^{2}}\,dx$
$\displaystyle=\frac{1}{2\mu}\int_{-2r}^{2r}(2\rho(x)\rho^{\prime}(x)x+\rho(x)^{2})e^{-\mu
x^{2}}\,dx$
$\displaystyle=\frac{1}{2\mu}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{1}{2}}+O(e^{-c\mu})\;.$
(3.33)

So

$\displaystyle\partial_{\mu}a_{\mu}$
$\displaystyle=\partial_{\mu}\bigg{(}\bigg{(}\int_{-2r}^{2r}\rho(x)^{2}e^{-\mu
x^{2}}\,dx\bigg{)}^{\frac{n}{2}}\bigg{)}$
$\displaystyle=-\frac{n}{2}\bigg{(}\int_{-2r}^{2r}\rho(x)^{2}e^{-\mu
x^{2}}\,dx\bigg{)}^{\frac{n}{2}-1}\int_{-2r}^{2r}\rho(x)^{2}x^{2}e^{-\mu
x^{2}}\,dx$
$\displaystyle=-\frac{n}{2}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{n}{4}-\frac{1}{2}}\frac{1}{2\mu}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{1}{2}}+O(e^{-c\mu})=-\frac{n}{4\mu}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{n}{4}}+O(e^{-c\mu})\;.$

Hence, by (3.12),

$\displaystyle\partial_{\mu}\Big{(}\frac{1}{a_{\mu}}\Big{)}$
$\displaystyle=-\frac{\partial_{\mu}a_{\mu}}{a_{\mu}^{2}}=\frac{n}{4\mu}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{n}{4}}\Big{(}\frac{\mu}{\pi}\Big{)}^{\frac{n}{2}}+O(e^{-c\mu})$
$\displaystyle=\frac{n}{4\mu}\Big{(}\frac{\mu}{\pi}\Big{)}^{\frac{n}{4}}+O(e^{-c\mu})\;.$
(3.34)

It also follows from Proposition 3.1 iii, (3.10), (3.12) and (3.34) that

$\displaystyle\partial_{\mu}e_{p,\mu}$
$\displaystyle=\partial_{\mu}\Big{(}\frac{\rho_{p}}{a_{\mu}}e^{-\mu|x_{p}|^{2}/2}\,dx_{p}^{1}\wedge\dots\wedge
dx_{p}^{\operatorname{ind}(p)}\Big{)}$
$\displaystyle=\bigg{(}\partial_{\mu}\Big{(}\frac{1}{a_{\mu}}\Big{)}a_{\mu}-\frac{|x_{p}|^{2}}{2}\bigg{)}e_{p,\mu}$
$\displaystyle=\bigg{(}\frac{n}{4\mu}-\frac{|x_{p}|^{2}}{2}+O(e^{-c\mu})\bigg{)}e_{p,\mu}\;.$
(3.35)

So, by (3.11),

$\partial_{\mu}e_{p,z}=\bigg{(}\frac{n}{4\mu}-\frac{|x_{p}|^{2}}{2}+O(e^{-c\mu})\bigg{)}e_{p,z}\;,\quad\partial_{\nu}e_{p,z}=-ih_{p}e_{p,z}\;.$
(3.36)

Then the result follows using (3.5). ∎

######  Proposition 3.24.

For all $p\in\mathcal{X}$,

$\|\partial_{z}(D_{z}e_{p,z})\|_{L^{\infty}}=O(e^{-c\mu})\quad(\mu\to+\infty)\;.$

###### Proof.

From (3.13), we get

$\partial_{z}(D_{z}e_{p,z})=\frac{1}{2}\bigg{(}e^{-i\nu
h_{p}}\partial_{\mu}\Big{(}\frac{1}{a_{\mu}}\Big{(}\frac{\pi}{\mu}\Big{)}^{n/4}\Big{)}\hat{c}(d\rho_{p})e_{p,\mu}\\\
{}+e^{-i\nu
h_{p}}\frac{1}{a_{\mu}}\Big{(}\frac{\pi}{\mu}\Big{)}^{n/4}\hat{c}(d\rho_{p})\partial_{\mu}e_{p,\mu}\\\
{}-h_{p}e^{-i\nu
h_{p}}\frac{1}{a_{\mu}}\Big{(}\frac{\pi}{\mu}\Big{)}^{n/4}\hat{c}(d\rho_{p})e_{p,\mu}\bigg{)}\;.$
(3.37)

By (3.12) and (3.34),

$\displaystyle\partial_{\mu}\Big{(}\frac{1}{a_{\mu}}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{n}{4}}\Big{)}$
$\displaystyle=\partial_{\mu}\Big{(}\frac{1}{a_{\mu}}\Big{)}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{n}{4}}-\frac{n\pi}{4a_{\mu}\mu^{2}}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{n}{4}-1}$
$\displaystyle=\frac{n}{4\mu}\Big{(}\frac{\mu}{\pi}\Big{)}^{\frac{n}{4}}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{n}{4}}-\frac{n\pi}{4\mu^{2}}\Big{(}\frac{\pi}{\mu}\Big{)}^{\frac{n}{4}-1}\Big{(}\frac{\mu}{\pi}\Big{)}^{\frac{n}{4}}+O(e^{-c\mu})$
$\displaystyle=O(e^{-c\mu})\;.$ (3.38)

The result follows applying Proposition 3.1 iii, (3.10), (3.12), (3.35) and
(3.38) to (3.37), and using that $d\rho_{p}=0$ around $p$. ∎

######  Proposition 3.25.

For every $p\in\mathcal{X}$,

$\|\partial_{z}(P_{z,\text{\rm
sm}}e_{p,z}-e_{p,z})\|_{L^{\infty}}=O(e^{-c\mu})\quad(\mu\to+\infty)\;.$

###### Proof.

By (3.18),

$\partial_{z}(P_{z,\text{\rm sm}}e_{p,z}-e_{p,z})=\frac{1}{2\pi
i}\int_{\mathbb{S}^{1}}w^{-1}\partial_{z}\big{(}(w-D_{z})^{-1}\big{)}D_{z}e_{p,z}\,dw\\\
+\frac{1}{2\pi
i}\int_{\mathbb{S}^{1}}w^{-1}(w-D_{z})^{-1}\partial_{z}(D_{z}e_{p,z})\,dw\;.$

Now apply (3.19), (3.30), Corollaries 3.6 and 3.24. ∎

##  4\. Zeta invariants of Morse forms

###  4.1. Preliminaries on asymptotic expansions of heat kernels

Let $A$ be a positive semi-definite symmetric elliptic differential operator
of order $a$, and $B$ a differential operator of order $b$; both of them are
defined in $C^{\infty}(M;E)$ for some Hermitian vector bundle $E$ over $M$.
Then $Be^{-tA}$ is a smoothing operator with Schwartz kernel $K_{t}(x,y)$ in
$C^{\infty}(M^{2};E\boxtimes E^{*})$ (omitting the Riemannian density
$\operatorname{vol}(y)$ of the second factor). On the diagonal, there is an
asymptotic expansion (as $t\downarrow 0$) [29, Lemma 1.9.1], [6, Theorem
2.30],

$K_{t}(x,x)\sim\sum_{l=0}^{\infty}e_{l}(x)t^{(l-n-b)/a}\;,$ (4.1)

where $e_{l}\in C^{\infty}(M;E\otimes E^{*})$ is locally computable in terms
of the jets of the local coefficients of $A$ and $B$. Hence the function

$h(t)=\operatorname{Tr}\big{(}Be^{-tA}\big{)}=\int_{M}\operatorname{tr}K_{t}(x,x)\,\operatorname{vol}(x)$

has an asymptotic expansion

$h(t)\sim\sum_{l=0}^{\infty}a_{l}t^{(l-n-b)/a}\;,$ (4.2)

where

$a_{l}=\int_{M}\operatorname{tr}e_{l}(x)\,\operatorname{vol}(x)\;.$ (4.3)

Given any $\lambda\geq 0$, let $P_{A,\lambda}$ be the spectral projection of
$A$ corresponding to $[0,\lambda]$; thus $P_{A,\lambda}^{\perp}$ is the
spectral projection corresponding to $(\lambda,\infty)$. By ellipticity,
$P_{A,\lambda}$ is of finite rank, and $Be^{-tA}P_{A,\lambda}$ is a smoothing
operator defined for all $t\in\mathbb{R}$. Take any orthonormal frame
$\phi_{1},\dots,\phi_{\kappa}$ of $\operatorname{im}P_{A,\lambda}$, consisting
of eigensections with corresponding eigenvalues
$0\leq\lambda_{1}\leq\dots\leq\lambda_{\kappa}\leq\lambda$. Then the Schwartz
kernel $H_{t}(x,y)$ of $Be^{-tA}P_{A,\lambda}$ ($t\geq 0$) is given by

$H_{t}(x,y)=\sum_{j=1}^{\kappa}e^{-t\lambda_{j}}(B\phi_{j})(x)\otimes\phi_{j}(y)\;,$

using the identity $E\equiv E^{*}$ given by the Hermitian structure. Thus
$H_{t}(x,y)$ is defined for all $t\in\mathbb{R}$ and smooth. So

$\operatorname{Tr}(Be^{-tA}P_{A,\lambda})=\int_{M}\operatorname{tr}H_{t}(x,x)\,\operatorname{vol}(x)\;.$

In particular, for $t=0$, we have

$\displaystyle H_{0}(x,x)$
$\displaystyle=\sum_{j=1}^{\kappa}(B\phi_{j})(x)\otimes\phi_{j}(x)\;,$ (4.4)
$\displaystyle\operatorname{Tr}(BP_{A,\lambda})$
$\displaystyle=\int_{M}\operatorname{tr}H_{0}(x,x)\,\operatorname{vol}(x)\;.$
(4.5)

The Schwartz kernel of $Be^{-tA}P_{A,\lambda}^{\perp}$ is
$\widetilde{K}_{t}(x,y)=K_{t}(x,y)-H_{t}(x,y)$ ($t>0$), which has an
asymptotic expansion

$\widetilde{K}_{t}(x,x)\sim\sum_{l=0}^{\infty}\tilde{e}_{l}(x)t^{(l-n-b)/a}\;,$
(4.6)

where the first $n+b$ sections $\tilde{e}_{l}$ are given by

$\tilde{e}_{l}(x)=\begin{cases}e_{l}(x)&\text{if $l<n+b$}\\\
e_{l}(x)-H_{0}(x,x)&\text{if $l=n+b$}\;.\end{cases}$

Then the function

$\tilde{h}_{\lambda}(t)=\operatorname{Tr}\big{(}Be^{-tA}P_{A,\lambda}^{\perp}\big{)}=\operatorname{Tr}\big{(}Be^{-tA}\big{)}-\operatorname{Tr}(Be^{-tA}P_{A,\lambda})$
(4.7)

has an asymptotic expansion

$\tilde{h}_{\lambda}(t)=\int_{M}\widetilde{K}_{t}(x,x)\,\operatorname{vol}(x)\sim\sum_{l=0}^{\infty}\tilde{a}_{\lambda,l}t^{(l-n-b)/a}\;,$
(4.8)

where the first $n+b$ coefficients $\tilde{a}_{l}$ are given by

$\tilde{a}_{\lambda,l}=\begin{cases}a_{l}&\text{if $l<n+b$}\\\
a_{l}-\operatorname{Tr}(BP_{A,\lambda})&\text{if $l=n+b$}\;.\end{cases}$ (4.9)

Consider also smooth families of such operators, $\\{A_{\epsilon}\\}$ and
$\\{B_{\epsilon}\\}$, for $\epsilon$ in some parameter space. Then
$\operatorname{Tr}(B_{\epsilon}e^{-tA_{\epsilon}})$ is smooth in
$(t,\epsilon)$, and we add $\epsilon$ to the above notation, writing for
instance $K_{t}(x,y,\epsilon)$, $e_{l}(x,\epsilon)$, $h(t,\epsilon)$,
$a_{l}(\epsilon)$, $\widetilde{K}_{t}(x,y,\epsilon)$,
$\tilde{e}_{\lambda,l}(x,\epsilon)$, $\tilde{h}_{\lambda}(t,\epsilon)$ and
$\tilde{a}_{\lambda,l}(\epsilon)$ in (4.1), (4.2), (4.6) and (4.8). The
operator $B_{\epsilon}P_{A_{\epsilon},\lambda}$ may not be smooth in
$\epsilon$ when some non-constant spectral branch of $\\{A_{\epsilon}\\}$
reaches the value $\lambda$. If the values of the non-constant spectral
branches of $\\{A_{\epsilon}\\}$ stay uniformly away from $\lambda$, then
$\tilde{h}_{\lambda}(t,\epsilon)$ is smooth in $(t,\epsilon)$.

###  4.2. Preliminaries on zeta functions of operators

######  Proposition 4.1 (See e.g. [29, Theorems 1.12.2 and 1.12.5], [6,
Propositions 9.35–9.37]).

The following holds:

  1. (i)

For every $\lambda\geq 0$, there is a meromorphic function
$\zeta(s,A,B,\lambda)$ on $\mathbb{C}$ such that, for $\Re s\gg 0$,

$\zeta(s,A,B,\lambda)=\operatorname{Tr}\big{(}BA^{-s}P_{A,\lambda}^{\perp}\big{)}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\tilde{h}_{\lambda}(t)\,dt\;.$
(4.10)

  2. (ii)

The meromorphic function $\Gamma(s)\zeta(s,A,B,\lambda)$ has simple poles at
the points $s=(n+b-l)/a$, for $l\in\mathbb{N}_{0}$ with $\tilde{a}_{l}\neq 0$.
The corresponding residues are $\tilde{a}_{l}$, and $\zeta(s,A,B,\lambda)$ is
smooth away from these exceptional values of $s$.

  3. (iii)

For $\mu>\lambda\geq 0$, let $\lambda_{1}\leq\dots\leq\lambda_{k}$ denote the
eigenvalues of $A$ in $(\lambda,\mu]$, taking multiplicities into account, and
let $\psi_{1},\dots,\psi_{k}$ be corresponding orthonormal eigensections.
Then, for all $s$,

$\zeta(s,A,B,\mu)-\zeta(s,A,B,\lambda)=\sum_{j=1}^{k}\lambda_{k}^{-s}\langle
B\psi_{j},\psi_{j}\rangle\;.$

  4. (iv)

For smooth families $\\{A_{\epsilon}\\}$ and $\\{B_{\epsilon}\\}$ of such
operators, if the non-constant branches of eigenvalues of $\\{A_{\epsilon}\\}$
stay uniformly away from the value $\lambda$, then
$\zeta(s,A_{\epsilon},B_{\epsilon},\lambda)$ is smooth in $(s,\epsilon)$ away
from the exceptional values of $s$ given in ii.

  5. (v)

Consider the conditions of iv for $\epsilon$ in some open neighborhood of $0$
in $\mathbb{R}$. If $A_{0}$ and $B_{0}$ commute, then

$\partial_{\epsilon}\zeta(s,A_{\epsilon},B_{\epsilon},\lambda)\big{|}_{\epsilon=0}=\zeta(s,A_{0},\dot{B}_{0},\lambda)-s\zeta(s+1,A_{0},\dot{A}_{0}B_{0},\lambda)\;,$

where the dot denotes $\partial_{\epsilon}$.

The last expression of (4.10) is the Mellin transform of the function
$\tilde{h}_{\lambda}(t)$ divided by $\Gamma(s)$. This function
$\zeta(s,A,B,\lambda)$ is called the _zeta function_ of $(A,B,\lambda)$. If
$B=1$ or $\lambda=0$, they may be omitted from the notation.

We will also use $\zeta(s,A,B,\lambda)$ when $B$ is not a differential
operator, with the same definition. Then the asymptotic expansion (4.8) and
the properties stated in Proposition 4.1 need to be checked. With this
generality, we can write

$\zeta(s,A,B,\lambda)=\zeta(s,A,BP_{A,\lambda}^{\perp})=\zeta(s,A,P_{A,\lambda}^{\perp}B)\;,$

and we have

$\zeta(s,A,B)=\zeta(s,A,BP_{A,\lambda})+\zeta(s,A,B,\lambda)\;.$

The function $\zeta(s,A,BP_{A,\lambda})$ is always defined and holomorphic on
$\mathbb{C}$ because $P_{A,\lambda}$ is of finite rank.

In particular, when $A$ is $\Delta_{z}$ or $\Delta_{z,k}$ ($k=0,\dots,n$) and
$B$ is another operator in $L^{2}(M;\Lambda)$, we will use
$\zeta(s,A,B,\lambda)$ if it is defined. We get

$\zeta(s,\Delta_{z},B)=\zeta_{\text{\rm sm}}(s,\Delta_{z},B)+\zeta_{\text{\rm
la}}(s,\Delta_{z},B)\;,$

where, with the notation of Section 3.2,

$\zeta_{\text{\rm sm/la}}(s,\Delta_{z},B)=\zeta(s,\Delta_{z},B_{z,\text{\rm
sm/la}})\;.$

These are the contributions from the small/large spectrum to
$\zeta(s,\Delta_{z},B)$, which are called the _small/large zeta functions_ of
$(\Delta_{z},B)$.

###  4.3. Zeta invariants of Morse forms

By Proposition 4.1 i, as a function of $s$,
$\zeta(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is meromorphic on
$\mathbb{C}$. Moreover, for $\Re s\gg 0$,

$\displaystyle\zeta(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$
$\displaystyle=\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}\Delta_{z}^{-s}\Pi_{z}^{\perp}\big{)}=\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,\delta_{z}\Delta_{z}^{-s}\Pi_{z}^{1}\big{)}$
$\displaystyle=\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}^{-1}\Delta_{z}^{-s+1}\Pi^{\perp}_{z}\big{)}=\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,d_{z}^{-1}\Delta_{z}^{-s+1}\Pi^{1}_{z}\big{)}\;,$

using that ${\eta\wedge}\,d_{z}$ and ${\eta\wedge}\,\delta_{z}^{-1}$ change
the degree of homogeneous forms. So, when
$\zeta(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is regular at $s=1$, the
value $\zeta(1,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is a renormalized
version of the super-trace of ${\eta\wedge}\,d_{z}^{-1}\Pi^{1}_{z}$, which is
called the _zeta invariant_ of $(M,g,\eta)$ for the scope of this paper.
Unfortunately, according to Proposition 4.1 ii and since $\Gamma(s)$ is
regular at $s=1$, $\zeta(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ might
have a simple pole at $s=1$. It will be shown that we can choose $\eta$ in the
given class $\xi\in H^{1}(M,\mathbb{R})$ such that
$\zeta(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is regular at $s=1$ for
$\mu\gg 0$. To achieve this task, we consider its decomposition into
small/large zeta functions (Section 4.2),

$\zeta(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})=\zeta_{\text{\rm
sm}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})+\zeta_{\text{\rm
la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})\;.$

The values $\zeta_{\text{\rm
sm/la}}(1,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ will be called the
_small/large zeta invariant_ of $(M,g,\eta)$, if they are
defined—$\zeta_{\text{\rm sm}}(1,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is
always defined.

###  4.4. Heat invariants of perturbed operators

For $k=0,\dots,n$, the Schwartz kernels of $e^{-t\Delta_{z,k}}$ and
$e^{-t\Delta_{z,k}}P_{z,\text{\rm la},k}$ are denoted by $K_{z,k,t}(x,y)$ and
$\widetilde{K}_{z,k,t}(x,y)$, respectively. Their restrictions to the diagonal
have asymptotic expansions (as $t\downarrow 0$),

$\displaystyle K_{z,k,t}(x,x)$
$\displaystyle\sim\sum_{l=0}^{\infty}e_{k,l}(x,z)t^{(l-n)/2}\;,$
$\displaystyle\widetilde{K}_{z,k,t}(x,x)$
$\displaystyle\sim\sum_{l=0}^{\infty}\tilde{e}_{k,l}(x,z)t^{(l-n)/2}\;,$
(4.11)

where $e_{k,l}(x,z)$ is locally given by smooth expressions involving $z$ and
the jets of the local coefficients of $g$ and $\eta$. Take $\mu\gg 0$ so that
$P_{z,\text{\rm la},k}$ depends smoothly on $z$ (Theorem 3.10) and $\dim
E_{z,\text{\rm sm}}^{k}=|\mathcal{X}_{k}|$ (Corollary 3.9). Thus
$\tilde{e}_{k,l}(x,z)$ also depends smoothly on $z$ (Section 4.1). Then

$\tilde{e}_{k,l}(x,z)=\begin{cases}e_{k,l}(x,z)&\text{if $l<n$}\\\
e_{k,n}(x,z)-H_{z,k,0}(x,x)&\text{if $l=n$}\;,\end{cases}$ (4.12)

where $H_{z,k,t}(x,y)$ is the Schwartz kernel of
$e^{-t\Delta_{z,k}}P_{z,\text{\rm sm},k}$, which is defined for all
$t\in\mathbb{R}$ and smooth. According to Section 4.2, the corresponding
functions

$h_{k}(t,z)=\operatorname{Tr}\big{(}e^{-t\Delta_{z,k}}\big{)}\;,\quad\tilde{h}_{k}(t,z)=\operatorname{Tr}\big{(}e^{-t\Delta_{z,k}}P_{z,\text{\rm
la},k}\big{)}\;,$

have asymptotic expansions

$h_{k}(t,z)\sim\sum_{l=0}^{\infty}a_{k,l}(z)t^{(l-n)/2}\;,\quad\tilde{h}_{k}(t,z)\sim\sum_{l=0}^{\infty}\tilde{a}_{k,l}(z)t^{(l-n)/2}\;,$
(4.13)

where, by (4.3)–(4.5) and (4.9),

$\displaystyle a_{k,l}(z)$ $\displaystyle=\int_{M}\operatorname{tr}^{\text{\rm
s}}e_{k,l}(x,z)\,\operatorname{vol}(x)\;,$ (4.14)
$\displaystyle\tilde{a}_{k,l}(z)$
$\displaystyle=\int_{M}\operatorname{tr}^{\text{\rm
s}}\tilde{e}_{k,l}(x,z)\,\operatorname{vol}(x)=\begin{cases}a_{k,l}(z)&\text{if
$l<n$}\\\ a_{k,l}(z)-|\mathcal{X}_{k}|&\text{if $l=n$}\;.\end{cases}$ (4.15)

We will also consider the operators $e^{-t\Delta_{z}}\mathsf{w}$ and
$e^{-t\Delta_{z}}P_{z,\text{\rm la}}\mathsf{w}$, whose respective Schwartz
kernels are

$K_{z,t}(x,y)=\sum_{k=0}^{n}(-1)^{k}K_{z,k,t}(x,y)\;,\quad\widetilde{K}_{z,t}(x,y)=\sum_{k=0}^{n}(-1)^{k}\widetilde{K}_{z,k,t}(x,y)\;.$

We have induced asymptotic expansions

$K_{z,t}(x,x)\sim\sum_{l=0}^{\infty}e_{l}(x,z)t^{(l-n)/2}\;,\quad\widetilde{K}_{z,t}(x,x)\sim\sum_{l=0}^{\infty}\tilde{e}_{l}(x,z)t^{(l-n)/2}\;,$

where

$e_{l}(x,z)=\sum_{k=0}^{n}(-1)^{k}e_{k,l}(x,z)\;,\quad\tilde{e}_{l}(x,z)=\sum_{k=0}^{n}(-1)^{k}\tilde{e}_{k,l}(x,z)\;.$

The corresponding functions,

$\displaystyle h(t,z)$ $\displaystyle=\operatorname{Tr}^{\text{\rm
s}}\big{(}e^{-t\Delta_{z}}\big{)}=\sum_{k=0}^{n}(-1)^{k}h_{k}(t,z)\;,$
$\displaystyle\tilde{h}(t,z)$ $\displaystyle=\operatorname{Tr}^{\text{\rm
s}}\big{(}e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}=\sum_{k=0}^{n}(-1)^{k}\tilde{h}_{k}(t,z)\;,$

have asymptotic expansions

$h(t,z)\sim\sum_{l=0}^{\infty}a_{l}(z)t^{(l-n)/2}\;,\quad\tilde{h}(t,z)\sim\sum_{l=0}^{\infty}\tilde{a}_{l}(z)t^{(l-n)/2}\;,$

where

$a_{l}(z)=\sum_{k=0}^{n}(-1)^{k}a_{k,l}(z)\;,\quad\tilde{a}_{l}(z)=\sum_{k=0}^{n}(-1)^{k}\tilde{a}_{k,l}(z)\;.$

######  Theorem 4.2 ([2, Theorem 1.5], [5]).

We have:

  1. (i)

$e_{l}(x,z)=0$ for $l<n$; and,

  2. (ii)

if $n$ is even, then $e_{n}(x,z)=e(M,\nabla^{M})(x)$.

######  Remark 4.3.

Actually, [2, Theorem 1.5] gives Theorem 4.2 when $z$ is real. But, since the
functions $e_{l}(x,z)$ have local expressions, we can assume $\eta$ is exact.
Then the result can be extended to non-real $z$ using (2.12).

###  4.5. Derived heat invariants of perturbed operators

For $k=0,\dots,n$ and $j=1,2$, let

$\displaystyle
h^{j}_{k}(t,z)=\operatorname{Tr}\big{(}e^{-t\Delta_{z,k}}\Pi^{j}_{z,k}\big{)}\;,\quad\tilde{h}^{j}_{k}(t,z)=\operatorname{Tr}\big{(}e^{-t\Delta_{z,k}}\Pi^{j}_{z,\text{\rm
la},k}\big{)}\;.$

######  Lemma 4.4.

We have

$h_{k+1}^{1}(t,z)=h_{k}^{2}(t,z)=\sum_{p=0}^{k}(-1)^{k-p}h_{p}(t,z)=\sum_{q=k+1}^{n}(-1)^{q-k-1}h_{q}(t,z)\;.$

###### Proof.

This follows by induction on $k$, using that

$\displaystyle h_{0}^{1}(t,z)=h_{n}^{2}(t,z)=0\;,\quad
h_{k}(t,z)=h_{k}^{1}(t,z)+h_{k}^{2}(t,z)\;,$ $\displaystyle
h_{k}^{2}(t,z)=h_{k+1}^{1}(t,z)\;.$

The last equality holds because (2.8) is commutative. ∎

Let

$\displaystyle h^{j}(t,z)$
$\displaystyle=\sum_{k=0}^{n}(-1)^{k}h^{j}_{k}(t,z)=\operatorname{Tr}^{\text{\rm
s}}\big{(}e^{-t\Delta_{z}}\Pi^{j}_{z}\big{)}\;,$
$\displaystyle\tilde{h}^{j}(t,z)$
$\displaystyle=\sum_{k=0}^{n}(-1)^{k}\tilde{h}^{j}_{k}(t,z)=\operatorname{Tr}^{\text{\rm
s}}\big{(}e^{-t\Delta_{z}}\Pi^{j}_{z,\text{\rm la}}\big{)}\;.$

Thus

$h(t,z)=h^{1}(t,z)+h^{2}(t,z)\;,\quad\tilde{h}(t,z)=\tilde{h}^{1}(t,z)+\tilde{h}^{2}(t,z)\;.$
(4.16)

######  Corollary 4.5.

We have $h(t,z)=0$.

###### Proof.

This is a direct consequence of Lemma 4.4 and (4.16). ∎

######  Corollary 4.6.

We have

$-(-1)^{j}h^{j}(t,z)=\sum_{k=0}^{n}(-1)^{k}kh_{k}(t,z)=\operatorname{Tr}^{\text{\rm
s}}\big{(}\mathsf{N}e^{-t\Delta_{z}}\Pi_{z}^{\perp}\big{)}\;.$

###### Proof.

By Lemmas 4.4 and 4.5,

$\displaystyle h^{1}(t,z)$
$\displaystyle=\sum_{k=0}^{n}(-1)^{k}\sum_{q=k}^{n}(-1)^{q-k}h_{q}(t,z)=\sum_{q=0}^{n}(-1)^{q}(q+1)h_{q}(t,z)$
$\displaystyle=h(t,z)+\sum_{q=0}^{n}(-1)^{q}qh_{q}(t,z)=\sum_{q=0}^{n}(-1)^{q}qh_{q}(t,z)\;.\qed$

Applying (4.13) and Lemma 4.4, we get

$h^{j}_{k}(t,z)\sim\sum_{l=0}^{\infty}a_{k,l}^{j}(z)t^{(l-n)/2}\;,\quad
h^{j}(t,z)\sim\sum_{l=0}^{\infty}a^{j}_{l}(z)t^{(l-n)/2}\;,$ (4.17)

where

$\displaystyle
a^{1}_{k+1,l}(z)=a^{2}_{k,l}(z)=\sum_{p=0}^{k}(-1)^{k-p}a_{p,l}(t,z)=\sum_{q=k+1}^{n}(-1)^{q-k-1}a_{q,l}(t,z)\;,$
$\displaystyle-(-1)^{j}a^{j}_{l}(z)=\sum_{k=0}^{n}(-1)^{k}ka_{k,l}(z)\;.$

This $a^{1}_{l}(z)$ is sometimes called the _derived heat invariant_ of
$\Delta_{z}$ [31], [59], [29, page 181], [1].

For $\mu\gg 0$, Lemma 4.4, Corollary 4.5 and (4.17) have obvious versions for
$\tilde{h}^{j}_{k}(t,z)$ and $\tilde{h}^{j}(t,z)$, with the similar proofs.
The coefficients of the corresponding asymptotic expansions are denoted by
$\tilde{a}^{j}_{k,l}(z)$ and $\tilde{a}^{j}_{l}(z)$.

######  Theorem 4.7 ([1, Theorem 1.3 (1b)]).

For all $l\leq n$, $a^{1}_{l}(z)$ and $\tilde{a}^{1}_{l}(z)$ are independent
of $z$.

######  Remark 4.8.

Again, [1, Theorem 1.3 (1b)] gives Theorem 4.7 in case of $a^{1}_{l}(z)$ for
real $z$, but this can be extended for non-real $z$ like in Remark 4.3. Then
the case of $\tilde{a}^{1}_{l}(z)$ follows from (4.15). In Theorem 4.7, the
case of $a^{1}_{l}(z)$ is true for any $\eta\in Z^{1}(M,\mathbb{R})$.

######  Remark 4.9.

Theorem 4.2 and the case of $a^{1}_{l}(z)$ in Theorem 4.7 hold for any
$\eta\in Z^{1}(M,\mathbb{R})$ (there is no need of other conditions). This is
also true for all properties of Sections 4.4 and 4.5 concerning
$K_{z,k,t}(x,y)$, $e_{k,l}(x,z)$, $h_{k}(t,z)$, $h(t,z)$, $h^{j}(t,z)$,
$h^{j}_{k}(t,z)$, $a_{k,l}(z)$, $a_{l}(z)$, $a^{j}_{k,l}(z)$,
$\zeta(s,\Delta_{z},\Pi_{z}^{j}\mathsf{w})$ and $\theta(s,z)$.

###  4.6. Regularity

By (4.8), we have an asymptotic expansion of the form

$\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}\sim\sum_{l=0}^{\infty}\tilde{b}_{l}(z)t^{(l-n-1)/2}\;.$ (4.18)

######  Lemma 4.10.

The function $\zeta_{\text{\rm
la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ has a simple pole at every
$s=(n+1-l)/2$ with $\tilde{b}_{l}(z)\neq 0$, for $1\leq l\leq n$ and for
even/odd $l\geq n+2$ if $n$ is even/odd, whose residue is $\tilde{b}_{l}(z)$,
and it is smooth away from these values of $s$. Moreover the value of
$\zeta_{\text{\rm la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ at every
regular point $s=(n+1-l)/2$, for odd $l\geq n+1$, is
$(l-n-1)!\,\tilde{b}_{l}(z)$.

###### Proof.

By Proposition 4.1 ii, the product $\Gamma(s)\zeta_{\text{\rm
la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ has a simple pole at every
$s=(n+1-l)/2$ with $\tilde{b}_{l}(z)\neq 0$ ($l\in\mathbb{N}_{0}$), whose
residue is $\tilde{b}_{l}(z)$, and $\zeta_{\text{\rm
la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is smooth away from these
exceptional values of $s$. Then the result follows because $\Gamma(s)$ has a
simple pole at every point $s=-k$ ($k\in\mathbb{N}_{0}$), whose residue is
$(-1)^{k}/k!$, and it is smooth on $\mathbb{C}\setminus(-\mathbb{N}_{0})$. ∎

By Corollary 4.6 and using the notation (1.4) and (1.5), we get

$\displaystyle(-1)^{j}\zeta(s,\Delta_{z},\Pi_{z}^{j}\mathsf{w})=\theta(s,z)\;,$
$\displaystyle(-1)^{j}\zeta_{\text{\rm
sm/la}}(s,\Delta_{z},\Pi_{z}^{j}\mathsf{w})=\theta_{\text{\rm sm/la}}(s,z)\;.$
(4.19)

Thus the following result follows like Lemma 4.10.

######  Lemma 4.11.

The function $\zeta_{\text{\rm la}}(s,\Delta_{z},\Pi_{z}^{1}\mathsf{w})$ has a
simple pole at every $s=(n-l)/2$ with $\tilde{a}^{1}_{l}(z)\neq 0$, for $1\leq
l\leq n-1$ and for odd/even $l\geq n+1$ if $n$ is even/odd, whose residue is
$\tilde{a}^{1}_{l}(z)$, and it is smooth away from these values of $s$.
Moreover the value of $\zeta_{\text{\rm
la}}(s,\Delta_{z},\Pi_{z}^{1}\mathsf{w})$ at every regular point $s=(n-l)/2$,
for even $l\geq n$, is $(l-n)!\,\tilde{a}^{1}_{l}(z)$.

According to Lemmas 4.10 and 4.11, $\zeta_{\text{\rm
la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ and $\zeta_{\text{\rm
la}}(s,\Delta_{z},\Pi_{z}^{1}\mathsf{w})$ are smooth at $s=0$ [64], but they
might have a simple pole at $s=1$.

######  Proposition 4.12.

We have

$\partial_{z}\zeta_{\text{\rm
la}}(s,\Delta_{z},\Pi^{1}_{z}\mathsf{w})=-s\zeta_{\text{\rm
la}}(s+1,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})\;.$

###### Proof.

Recall that a dot may be used to denote $\partial_{z}$. Like in (3.32),

$\dot{\Pi}^{1}_{z,\text{\rm la}}=\big{(}\Pi^{1}_{z,\text{\rm
la}}\big{)}^{\perp}\dot{\Pi}^{1}_{z,\text{\rm la}}\Pi^{1}_{z,\text{\rm
la}}+\Pi^{1}_{z,\text{\rm la}}\dot{\Pi}^{1}_{z,\text{\rm
la}}\big{(}\Pi^{1}_{z,\text{\rm la}}\big{)}^{\perp}\;.$ (4.20)

Therefore, for $\Re s\gg 0$,

$\zeta_{\text{\rm
la}}(s,\Delta_{z},\dot{\Pi}^{1}_{z}\mathsf{w})=\operatorname{Tr}^{\text{\rm
s}}\big{(}\dot{\Pi}^{1}_{z,\text{\rm la}}\Delta_{z}^{-s}\Pi^{1}_{z,\text{\rm
la}}\big{)}=0\;,$

yielding $\zeta_{\text{\rm la}}(s,\Delta_{z},\dot{\Pi}^{1}_{z}\mathsf{w})=0$
for all $s$ because this is a meromorphic function of $s$. Hence, since
$\Delta_{z}$ and $\Pi^{1}_{z,\text{\rm la}}\mathsf{w}$ commute, Proposition
4.1 i,v give

$\displaystyle\partial_{z}\zeta_{\text{\rm
la}}(s,\Delta_{z},\Pi^{1}_{z}\mathsf{w})$ $\displaystyle=-s\zeta_{\text{\rm
la}}(s+1,\Delta_{z},\dot{\Delta}_{z}\Pi^{1}_{z}\mathsf{w})$
$\displaystyle=-s\operatorname{Tr}^{\text{\rm
s}}\big{(}\dot{\Delta}_{z}\Delta_{z}^{-s-1}\Pi^{1}_{z,\text{\rm
la}}\big{)}\;.$ (4.21)

Next, by (2.4),

$\dot{\Delta}_{z}\Pi^{1}_{z,\text{\rm
la}}=({\eta\wedge}\,\delta_{z}+\delta_{z}\,{\eta\wedge})\Pi^{1}_{z,\text{\rm
la}}={\eta\wedge}\,\delta_{z}\Pi^{1}_{z,\text{\rm
la}}+\delta_{z}\,{\eta\wedge}\,\Pi^{1}_{z,\text{\rm la}}\;.$ (4.22)

But, since $\Pi^{1}_{z}\delta_{z}=0$,

$\operatorname{Tr}^{\text{\rm
s}}\big{(}\delta_{z}\,{\eta\wedge}\,\Delta_{z}^{-s-1}\Pi^{1}_{z,\text{\rm
la}}\big{)}=-\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,\Delta_{z}^{-s-1}\Pi^{1}_{z,\text{\rm
la}}\delta_{z}\big{)}=0\;.$ (4.23)

Combining (4.21)–(4.23) and Proposition 4.1 i, we get

$\displaystyle\partial_{z}\zeta_{\text{\rm
la}}(s,\Delta_{z},\Pi^{1}_{z}\mathsf{w})$
$\displaystyle=-s\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,\delta_{z}\Delta_{z}^{-s-1}\Pi^{1}_{z,\text{\rm
la}}\big{)}$ $\displaystyle=-s\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}\Delta_{z}^{-s-1}\Pi^{1}_{z,\text{\rm
la}}\big{)}$ $\displaystyle=-s\zeta_{\text{\rm
la}}(s+1,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})\;.\qed$

######  Theorem 4.13.

For $\mu\gg 0$, $\zeta_{\text{\rm
la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is smooth on
$\mathbb{C}\setminus((1-\mathbb{N}_{0})/2)$.

###### Proof.

By Lemma 4.10, it is enough to prove that $\zeta_{\text{\rm
la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is regular at
$s=s_{l}=(n+1-l)/2$ for $l\leq n-1$, which means $\tilde{b}_{l}(z)=0$.

First, take $l\leq n-2$. By Lemmas 4.10, 4.11, 4.7 and 4.12, for $\mu\gg 0$,

$0=\partial_{z}\tilde{a}^{1}_{l+1}(z)=-(s_{l}-1)\tilde{b}_{l}(z)\;.$

Thus $\tilde{b}_{l}(z)=0$ because $s_{l}-1=(n-1-l)/2>0$.

Next, take $l=n-1$. (The previous argument does not apply in this case because
$s_{n-1}-1=0$.) By Theorems 4.7 and 4.11, for $\mu\gg 0$,

$\partial_{z}\zeta_{\text{\rm
la}}(0,\Delta_{z},\Pi^{1}_{z}\mathsf{w})=\partial_{z}\tilde{a}^{1}_{n}(z)=0\;.$

So there is a holomorphic function $\phi_{z}(s)$, defined around $s=0$ and
depending smoothly on $z$, such that

$\partial_{z}\zeta_{\text{\rm
la}}(s,\Delta_{z},\Pi^{1}_{z}\mathsf{w})=s\phi_{z}(s)\;.$

Hence, by Proposition 4.12,

$\phi_{z}(s)=-\zeta_{\text{\rm
la}}(s+1,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})\;,$

obtaining that $\zeta_{\text{\rm
la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})$ is regular at $s=s_{n-1}=1$.
∎

From now on, we use the simpler notation

$\zeta(s,z)=\zeta(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})\;,\quad\zeta_{\text{\rm
sm/la}}(s,z)=\zeta_{\text{\rm
sm/la}}(s,\Delta_{z},{\eta\wedge}\,D_{z}\mathsf{w})\;.$

We may also add $\eta$ to this notation, writing $\zeta(s,z,\eta)$ and
$\zeta_{\text{\rm sm/la}}(s,z,\eta)$.

######  Corollary 4.14.

If $\Re s>1/2$ and $\mu\gg 0$, then

$\zeta_{\text{\rm
la}}(s,z)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}\,dt\;,$

where the integral is absolutely convergent.

###### Proof.

By Lemma 4.10, Theorem 4.13 and (4.18), for $\mu\gg 0$,

$\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}=O\big{(}t^{-1/2}\big{)}\quad(t\to 0)\;.$ (4.24)

Moreover there is some $c>0$ such that

$\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}=O(e^{-ct})\quad(t\to\infty)\;.$ (4.25)

So the stated integral is absolutely convergent for $\Re s>1/2$, defining a
holomorphic function of $s$ on this half-plane. Then the stated equality is
true because it holds for $\Re s\gg 0$. ∎

######  Corollary 4.15.

For $\mu\gg 0$,

$\displaystyle\zeta_{\text{\rm sm}}(1,z)$
$\displaystyle=\operatorname{Tr}^{\text{\rm
s}}({\eta\wedge}\,D_{z}^{-1}(\Pi_{z}^{\perp})_{\text{\rm sm}})\;,$
$\displaystyle\zeta_{\text{\rm la}}(1,z)$ $\displaystyle=\lim_{t\downarrow
0}\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}^{-1}e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}\;,$ $\displaystyle\zeta(1,z)$ $\displaystyle=\lim_{t\downarrow
0}\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}^{-1}e^{-t\Delta_{z}}\Pi^{\perp}_{z}\big{)}\;.$

###### Proof.

By Corollary 4.14, (4.24) and (4.25), and since

$\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}^{-1}e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}=O(e^{-ct})\quad(t\to\infty)\;,$

we get

$\displaystyle\zeta_{\text{\rm la}}(1,z)$
$\displaystyle=\int_{0}^{\infty}\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}e^{-u\Delta_{z}}P_{z,\text{\rm la}}\big{)}\,du$
$\displaystyle=\lim_{t\downarrow
0}\int_{t}^{\infty}\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}e^{-u\Delta_{z}}P_{z,\text{\rm la}}\big{)}\,du$
$\displaystyle=\lim_{t\downarrow 0}\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,D_{z}^{-1}e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}\;.$

The expressions of $\zeta(1,z)$ and $\zeta_{\text{\rm sm}}(1,z)$ follow from
the expression of $\zeta_{\text{\rm la}}(1,z)$ and Proposition 4.1 iii. ∎

Theorems 4.13 and 4.15 give Theorem 1.1 i.

######  Corollary 4.16.

If $\mu\gg 0$, then

$\zeta_{\text{\rm la}}(1,z)=-\partial_{z}\theta_{\text{\rm
la}}^{\prime}(0,z)\;.$

###### Proof.

ApplyPropositions 4.12 and 4.13. ∎

###  4.7. The case of exact forms

Let us consider the special case where $\eta=dh$ for a Morse function $h$.
These results will be extended later, but the arguments are much simpler in
this case, with weaker conditions and less ingredients.

######  Lemma 4.17.

For $\mu\gg 0$,

$\displaystyle\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,d_{z}^{-1}\Pi^{1}_{z,\text{\rm sm}}\big{)}$
$\displaystyle=-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,(\Pi_{z}^{\perp})_{\text{\rm sm}}\big{)}\;,$
$\displaystyle\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,d_{z}^{-1}e^{-t\Delta_{z}}\Pi^{1}_{z,\text{\rm
la}}\big{)}$ $\displaystyle=-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,e^{-t\Delta_{z}}P_{z,\text{\rm la}}\big{)}\;,$
$\displaystyle\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,d_{z}^{-1}e^{-t\Delta_{z}}\Pi^{1}_{z}\big{)}$
$\displaystyle=-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,e^{-t\Delta_{z}}\Pi_{z}^{\perp}\big{)}\;.$

###### Proof.

We prove the first equality, the other ones being analogous. Since
${\eta\wedge}=[d,h]$,

$\displaystyle\operatorname{Tr}^{\text{\rm
s}}\big{(}{\eta\wedge}\,d_{z}^{-1}\Pi^{1}_{z,\text{\rm sm}}\big{)}$
$\displaystyle=\operatorname{Tr}^{\text{\rm
s}}\big{(}[d_{z},h]\,d_{z}^{-1}\Pi^{1}_{z,\text{\rm sm}}\big{)}$
$\displaystyle=\operatorname{Tr}^{\text{\rm
s}}\big{(}d_{z}\,h\,d_{z}^{-1}\Pi^{1}_{z,\text{\rm
sm}}\big{)}-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,d_{z}d_{z}^{-1}\Pi^{1}_{z,\text{\rm sm}}\big{)}$
$\displaystyle=-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,d_{z}^{-1}\Pi^{1}_{z,\text{\rm
sm}}d_{z}\big{)}-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,\Pi^{1}_{z,\text{\rm sm}}\big{)}$
$\displaystyle=-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,d_{z}^{-1}d_{z}\Pi^{2}_{z,\text{\rm
sm}}\big{)}-\operatorname{Tr}^{\text{\rm s}}\big{(}h\,\Pi^{1}_{z,\text{\rm
sm}}\big{)}$ $\displaystyle=-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,\Pi^{2}_{z,\text{\rm sm}}\big{)}-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,\Pi^{1}_{z,\text{\rm sm}}\big{)}$
$\displaystyle=-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,(\Pi_{z}^{\perp})_{\text{\rm sm}}\big{)}\;.\qed$

######  Remark 4.18.

The last equality of Lemma 4.17 does not require any condition on $\eta$ or
$g$ ($h$ can be any smooth real-valued function).

######  Corollary 4.19.

For $\mu\gg 0$,

$\displaystyle\zeta_{\text{\rm sm}}(1,z)$
$\displaystyle=-\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,(\Pi_{z}^{\perp})_{\text{\rm sm}}\big{)}\;,$
$\displaystyle\zeta_{\text{\rm la}}(1,z)$ $\displaystyle=-\lim_{t\downarrow
0}\operatorname{Tr}^{\text{\rm s}}\big{(}h\,e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}\;,$ $\displaystyle\zeta(1,z)$ $\displaystyle=-\lim_{t\downarrow
0}\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,e^{-t\Delta_{z}}\Pi_{z}^{\perp}\big{)}\;.$

###### Proof.

Apply Corollaries 4.15 and 4.17. ∎

######  Corollary 4.20.

The value $\zeta_{\text{\rm sm}}(1,z)$ is uniformly bounded on $z$ for $\mu\gg
0$.

###### Proof.

The operator $h\,(\Pi_{z}^{\perp})_{\text{\rm sm}}$ is uniformly bounded and,
for $\mu\gg 0$, has uniformly bounded rank. So $\operatorname{Tr}^{\text{\rm
s}}(h\,(\Pi_{z}^{\perp})_{\text{\rm sm}})$ is uniformly bounded on $z$ for
$\mu\gg 0$, and therefore the result follows from Corollary 4.19. ∎

######  Corollary 4.21.

If $\mu\gg 0$, then $\zeta(1,z),\zeta_{\text{\rm sm/la}}(1,z)\in\mathbb{R}$.

###### Proof.

We consider the case of $\zeta(1,z)$, the other cases being similar. By
Corollary 4.19, it is enough to prove that $\operatorname{Tr}^{\text{\rm
s}}(h\,e^{-t\Delta_{z}}\Pi_{z}^{\perp})\in\mathbb{R}$. This is true because,
taking adjoints,

$\displaystyle\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,e^{-t\Delta_{z}}\Pi_{z}^{\perp}\big{)}$
$\displaystyle=\overline{\operatorname{Tr}^{\text{\rm
s}}\big{(}\Pi_{z}^{\perp}e^{-t\Delta_{z}}\,h\big{)}}=\overline{\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,\Pi_{z}^{\perp}e^{-t\Delta_{z}}\big{)}}$
$\displaystyle=\overline{\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,e^{-t\Delta_{z}}\Pi_{z}^{\perp}\big{)}}\;.\qed$

######  Corollary 4.22.

If $M$ is oriented, $n$ is even and $|\mu|\gg 0$, then

$\displaystyle\zeta(1,z)=\zeta(1,-\bar{z})=\zeta(1,-z)=\zeta(1,\bar{z})\;,$
$\displaystyle\zeta_{\text{\rm sm/la}}(1,z)=\zeta_{\text{\rm
sm/la}}(1,-\bar{z})=\zeta_{\text{\rm sm/la}}(1,-z)=\zeta_{\text{\rm
sm/la}}(1,\bar{z})\;.$

###### Proof.

We prove the case of $\zeta_{\text{\rm sm}}(1,z)$, the other cases being
similar. By (2.16) and since $n$ is even,

$\displaystyle\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,(\Pi_{z}^{\perp})_{\text{\rm sm}}\big{)}$
$\displaystyle=\operatorname{Tr}^{\text{\rm
s}}\big{(}\star\star^{-1}h\,(\Pi_{z}^{\perp})_{\text{\rm
sm}}\big{)}=\operatorname{Tr}^{\text{\rm
s}}\big{(}\star^{-1}h\,(\Pi_{z}^{\perp})_{\text{\rm sm}}\star\big{)}$
$\displaystyle=\operatorname{Tr}^{\text{\rm s}}\big{(}\star^{-1}\star
h\,(\Pi_{-\bar{z}}^{\perp})_{\text{\rm
sm}}\big{)}=\operatorname{Tr}^{\text{\rm
s}}\big{(}h\,(\Pi_{-\bar{z}}^{\perp})_{\text{\rm sm}}\big{)}\;.$

Thus the first equality of the statement follows by Corollary 4.19. The second
equality follows with a similar argument, using $\bar{\star}$ instead of
$\star$. The third equality is a consequence of the other ones. ∎

######  Theorem 4.23.

The following limit holds uniformly on $\nu$:

$\lim_{\mu\to+\infty}\zeta_{\text{\rm
la}}(1,z)=-\int_{M}h\,e(M,\nabla^{M})\,\operatorname{vol}+\sum_{p\in\mathcal{X}}^{n}(-1)^{\operatorname{ind}(p)}h(p)\;.$

###### Proof.

By (4.11), (4.12), Theorems 4.2 and 4.19, for $\mu\gg 0$,

$\displaystyle\zeta_{\text{\rm la}}(1,z)$ $\displaystyle=-\lim_{t\downarrow
0}\operatorname{Tr}^{\text{\rm s}}\big{(}h\,e^{-t\Delta_{z}}P_{z,\text{\rm
la}}\big{)}$ $\displaystyle=-\int_{M}h(x)\,\operatorname{tr}^{\text{\rm
s}}\tilde{e}_{n}(x,z)\,\operatorname{vol}(x)$
$\displaystyle=-\int_{M}h(x)\,\operatorname{tr}^{\text{\rm
s}}e_{n}(x,z)\,\operatorname{vol}(x)+\operatorname{Tr}^{\text{\rm
s}}(hP_{z,\text{\rm sm}})$
$\displaystyle=-\int_{M}h\,e(M,\nabla^{M})\,\operatorname{vol}+\operatorname{Tr}^{\text{\rm
s}}(hP_{z,\text{\rm sm}})\;.$

According to Corollary 3.9, the elements $P_{z,\text{\rm sm}}e_{p,z}$
($p\in\mathcal{X}$) form a base of $E_{z,\text{\rm sm}}^{k}$ when $\mu\gg 0$.
Applying the Gram-Schmidt process to this base, we get an orthonormal frame
$\tilde{e}_{p,z}$ ($p\in\mathcal{X}$) of $E_{z,\text{\rm sm}}$. By Proposition
3.7 for $m=0$ and (3.9)–(3.12), we get

$\lim_{\mu\to+\infty}\langle
h\,\tilde{e}_{p,z},\tilde{e}_{q,z}\rangle=\lim_{\mu\to+\infty}\langle
he_{p,z},e_{q,z}\rangle=h(p)\delta_{pq}\;.$

Hence

$\lim_{\mu\to+\infty}\operatorname{Tr}^{\text{\rm s}}(hP_{z,\text{\rm
sm}})=\sum_{k=0}^{n}(-1)^{k}\sum_{p\in\mathcal{X}_{k}}h(p)\;.\qed$

##  5\. The small complex vs the Morse complex

###  5.1. Preliminaries on Smale vector fields

####  5.1.1. Vector fields with Morse type zeros

Let $X$ be a real smooth vector field on $M$ with flow $\phi=\\{\phi^{t}\\}$.
Let $\mathcal{X}=\operatorname{Zero}(X)$ denote the set of zeros of $X$ (or
rest points $\phi$). It is said that a zero $p$ of $X$ is of _Morse type_ with
(_Morse_) _index_ of $\operatorname{ind}(p)$ if, using the notation (3.2),

$X=-\sum_{j=1}^{n}\epsilon_{p,j}x_{p}^{j}\,\frac{\partial}{\partial
x_{p}^{j}}$ (5.1)

on the domain $U_{p}$ of some coordinates $x_{p}=(x_{p}^{1},\dots,x_{p}^{n})$
at $p$, also called _Morse coordinates_. This condition means that
$X=-\operatorname{grad}_{g}h_{X,p}$ on $U_{p}$, where $h_{X,p}$ and $g$ are
given on $U_{p}$ by the right-hand side of (3.1) and (3.3). The coordinates
$x_{p}$ used in (5.1) are not unique; that expression is invariant by taking
positive multiples of the coordinates (contrary to the expressions (3.1),
(3.3) and (3.4)). But $\operatorname{ind}(p)$ is independent of $x_{p}$. If
(3.3), (3.4) and (5.1) hold with the same coordinates, then $\eta$ and $g$ are
said to be in _standard form_ with respect to $X$ around $p$. In this case,
$C\eta$ and $Cg$ ($C>0$) are also in standard form with respect to $X$ around
$p$; indeed, $C\eta$, $X$ and $Cg$ satisfy (3.3), (3.4) and (5.1) with the
coordinates $\sqrt{C}x_{p}$. When $g$ is defined on $M$, if $\eta$ and $g$ are
in standard form with respect to $X$ around every $p\in\mathcal{X}$, then
$\eta$ and $g$ are said to be in _standard form_ with respect to $X$. This
concept is also applied to any Morse function $h$ on $M$ referring to $dh$ and
$g$. The reference to $g$ may be omitted in this terminology.

The vector field $X$ is fixed in most of the paper and plays an auxiliary
role. Unless otherwise indicated, we assume from now on that $X$ has Morse
type zeros. Then $\mathcal{X}$ is finite, and the sets $\mathcal{X}_{k}$,
$\mathcal{X}_{+}$ and $\mathcal{X}_{<k}$ are defined like in Section 3.1.

####  5.1.2. Stable and unstable manifolds

For $k=0,\dots,n$ and $p\in\mathcal{X}_{k}$, the _stable/unstable manifolds_
of $p$ are smooth injective immersions, $\iota^{\pm}_{p}:W^{\pm}_{p}\to M$,
where the images $\iota^{\pm}_{p}(W^{\pm}_{p})$ consist of the points
satisfying $\phi^{t}(x)\to p$ as $t\to\pm\infty$, and the manifolds
$W^{+}_{p}$ and $W^{-}_{p}$ are diffeomorphic to $\mathbb{R}^{n-k}$ and
$\mathbb{R}^{k}$, respectively [68, Theorem 9.1]. In particular,
$p\in\iota^{\pm}_{p}(W^{\pm}_{p})$, and the maps $\iota^{+}_{p}$ and
$\iota^{-}_{p}$ meet transversely at $p$. Let
$p^{\pm}=(\iota^{\pm}_{p})^{-1}(p)$. Assume every $U_{p}$ is connected, and
let $U^{\pm}_{p}$ be the connected component of
$(\iota^{\pm}_{p})^{-1}(U_{p})$ that contains $p^{\pm}$. The restriction
$\iota^{\pm}_{p}:U^{\pm}_{p}\to(x_{p}^{\pm})^{-1}(0)$ is a diffeomorphism, and
therefore $(U^{\pm}_{p},x^{\pm}_{p}\iota^{\pm}_{p})$ is a coordinate system of
$W^{\pm}_{p}$ centered at $p^{\pm}$.

####  5.1.3. Gradient-like vector fields

Given a Morse function $h$ on $M$ in standard form with respect to $X$, we
have $X=-\operatorname{grad}_{g}h$ on $M$ for some Riemannian metric $g$ if
and only if $Xh<0$ on $M\setminus\mathcal{X}$ [14, Lemma 2.1], [38, Section
6.1.3]; in this case, $X$ is said to be _gradient-like_ (with respect to $h$).
If $X$ is gradient-like, then the maps $\iota^{\pm}_{p}$ are embeddings [66,
Lemma 3.8], [14, Lemma 2.2], and their images cover $M$ [67, Lemma 1.1], [14,
Corollary 2.5]. Thus, in this case, we will write
$W^{\pm}_{p}=\iota^{\pm}_{p}(W^{\pm}_{p})$ and $p^{\pm}=p$, and
$\iota^{\pm}_{p}$ becomes the inclusion map.

Unless otherwise indicated, we also assume in the rest of the paper that $X$
is gradient-like.

####  5.1.4. Smale vector fields

$X$ is said to be _Smale_ if $W^{+}_{p}\pitchfork W^{-}_{q}$ for all
$p,q\in\mathcal{X}$. In this case, $\mathcal{M}(p,q):=W^{+}_{p}\cap W^{-}_{q}$
is a $\phi$-saturated smooth submanifold of dimension
$\operatorname{ind}(p)-\operatorname{ind}(q)$. In particular,
$\mathcal{M}(p,p)=\\{p\\}$, and define $\mathcal{T}(p,p)=\emptyset$. If $p\neq
q$, then the induced $\mathbb{R}$-action on $\mathcal{M}(p,q)$ is free and
proper, and therefore the orbit space
$\mathcal{T}(p,q):=\mathcal{M}(p,q)/\mathbb{R}$ is a smooth manifold of
dimension $\operatorname{ind}(p)-\operatorname{ind}(q)-1$. The elements of
$\mathcal{T}(p,q)$ are the (unparameterized) trajectories with $\alpha$-limit
$\\{p\\}$ and $\omega$-limit $\\{q\\}$, which are oriented by $X$. If
$\operatorname{ind}(p)\leq\operatorname{ind}(q)$, then
$\mathcal{T}(p,q)=\emptyset$. If
$\operatorname{ind}(p)-\operatorname{ind}(q)=1$, then $\mathcal{T}(p,q)$
consists of isolated points, each of them representing a trajectory in $M$.
Let $\mathcal{T}=\bigcup_{p,q\in\mathcal{X}}\mathcal{T}(p,q)$, and

$\mathcal{T}^{1}_{p}=\bigcup_{q\in\mathcal{X}_{\operatorname{ind}(p)-1}}\mathcal{T}(p,q)\;,\quad\mathcal{T}^{1}_{k}=\bigcup_{p\in\mathcal{X}_{k}}\mathcal{T}^{1}_{p}\;,\quad\mathcal{T}^{1}=\bigcup_{k=0}^{n}\mathcal{T}^{1}_{k}\;.$

The elements of $\mathcal{T}^{1}$ are called _instantons_.111Sometimes, the
elements of $\mathcal{T}$ are called _instantons_ , and the elements of
$\mathcal{T}^{1}$ _proper instantons_ [11].

Unless otherwise indicated, besides the above conditions, we assume from now
on that $X$ is Smale; i.e., we assume b (Section 1.1). Thus the $\alpha$\- and
$\omega$-limits of the orbits of $X$ are zero points [67, Theorem B and Lemma
1.1].

####  5.1.5. Lyapunov forms

Any $\eta\in Z^{1}(M,\mathbb{R})$ is said to be _Lyapunov_ for $X$ if
$\eta(X)<0$ on $M\setminus\mathcal{X}$ [18, Definition 2.3]. By b, every class
in $H^{1}(M,\mathbb{R})$ has a representative $\eta$ which is Lyapunov for $X$
and $\eta^{\sharp}=-X$ for some Riemannian metric $g$ on $M$, with $\eta$ and
$g$ in standard form with respect to $X$ [16, Proposition 16 (i)], [18,
Observations 2.5 and 2.6], [32, Lemma 3.7], [38, Section 6.1.3]. Moreover $X$
can be $C^{\infty}$-approximated by gradient-like Smale vector fields that
agree with $X$ around $\mathcal{X}$ [18, Proposition 2.4] (this follows from
[67, Theorem A]). A well known consequence is that, for any Morse function
$h$, there is a $C^{\infty}$-dense set of Riemannian metrics $g$ on $M$ such
that $-\operatorname{grad}_{g}h$ is Smale; this density is also true in the
subspace of metrics that are Euclidean with respect to Morse coordinates on
given neighborhoods of the critical points.

####  5.1.6. Completion of the unstable manifolds

######  Proposition 5.1 ([9, Appendix by F. Laudenbach, Proposition 2], [37,
Chapter 2], [13, Theorem 2.1], [15, Theorem 1], [14, Theorem 4.4], [38,
Sections A.2 and A.8], [49, Corollary 2.3.2]).

The following holds for every $p\in\mathcal{X}_{k}\ (k=0,\dots,n)$:

  1. (i)

$\overline{W^{-}_{p}}$ is a $C^{1}$ submanifold with conic singularities (in
the sense of [9, Appendix by F. Laudenbach, Section a)] and [38, Appendix
A.1]) and a Whitney stratified subspace. Its strata are the submanifolds
$W^{-}_{q}$ for $q\in\mathcal{X}_{<k}$ with $\mathcal{T}(p,q)\neq\emptyset$.
As a consequence, $W^{-}_{p}$ has finite volume, and

$\overline{W^{-}_{q}}\cap\overline{W^{-}_{p}}\subset\bigcup_{x\in\mathcal{X}_{<k}}W_{x}^{-}$

if $q\neq p$ in $\mathcal{X}_{k}$; in particular,
$p\notin\overline{W^{-}_{q}}$.

  2. (ii)

There is a compact $k$-manifold with corners222In the sense of [44, Section
1.1.8]. $\widehat{W}^{-}_{p}$ whose $l$-corner333The union of the interiors of
the boundary faces of codimension $l$. $(0\leq l\leq k)$ is

$\partial_{l}\widehat{W}^{-}_{p}=\bigsqcup_{(q_{0},\dots,q_{l})\in\\{p\\}\times\mathcal{X}^{l}}\bigg{(}\prod_{j=1}^{l}\mathcal{T}(q_{j-1},q_{j})\bigg{)}\times
W_{q_{l}}^{-}\;.$

In particular, the interior of $\widehat{W}^{-}_{p}$ is
$\partial_{0}\widehat{W}^{-}_{p}=W^{-}_{p}$, and the set $\mathcal{T}(p,q)$ is
finite if $q\in\mathcal{X}_{k-1}$.

  3. (iii)

There is a smooth map $\hat{\iota}^{-}_{p}:\widehat{W}^{-}_{p}\to M$ whose
restriction to every component of $\partial_{l}\widehat{W}^{-}_{p}$ in ii is
given by the factor projection to $W_{q_{l}}^{-}$; in particular,
$\hat{\iota}^{-}_{p}=\iota^{-}_{p}$ on $W^{-}_{p}$. Moreover
$\hat{\iota}_{p}^{-}:\widehat{W}^{-}_{p}\to\overline{W^{-}_{p}}$ is a
stratified map.

By Proposition 5.1 i, we can choose the open sets $U_{p}$
($p\in\mathcal{X}_{k}$, $k=0,\dots,n$) so small that
$U_{p}\cap\overline{W^{-}_{q}}=\emptyset$ if $q\neq p$ in $\mathcal{X}_{k}$.

For every $q\in\mathcal{X}_{k-1}$ and $\gamma\in\mathcal{T}(p,q)$, the closure
$\bar{\gamma}$ in $M$ is a compact oriented submanifold with boundary of
dimension one, and $\partial\bar{\gamma}=\\{p,q\\}$. We may also consider
$\bar{\gamma}$ as the closure of $\gamma$ in $\widehat{W}^{-}_{p}$.

###  5.2. Preliminaries on the Morse complex

####  5.2.1. The Morse complex

Fix an orientation $\mathcal{O}_{p}^{-}$ of every unstable manifold
$W^{-}_{p}$ ($p\in\mathcal{X}_{k}$, $k=0,\dots,n$), which can be also
considered as an orientation of $\widehat{W}^{-}_{p}$. Then
$W^{-}_{p}\equiv(W^{-}_{p},\mathcal{O}^{-}_{p})$ defines a current of
dimension $k$ on $M$, also denoted by $W^{-}_{p}$; namely, for
$\alpha\in\Omega^{k}(M)$,

$\langle
W^{-}_{p},\alpha\rangle=\int_{W^{-}_{p}}\alpha=\int_{\widehat{W}^{-}_{p}}(\hat{\iota}_{p}^{-})^{*}\alpha\;.$

Let $\partial_{1}\mathcal{O}_{p}^{-}$ be the orientation of
$\partial_{1}\widehat{W}^{-}_{p}$ induced by $\mathcal{O}_{p}^{-}$ like in the
Stokes’ theorem. The restriction of $\partial_{1}\mathcal{O}_{p}^{-}$ to every
component $\mathcal{T}(p,q)\times W^{-}_{q}$ ($q\in\mathcal{X}_{k^{\prime}}$)
of $\partial_{1}\widehat{W}^{-}_{p}$ is of the form
$\mathcal{O}_{p,q}\otimes\mathcal{O}_{q}^{-}$ for a unique orientation
$\mathcal{O}_{p,q}$ of $\mathcal{T}(p,q)$. If $k^{\prime}=k-1$, then
$\mathcal{O}_{p,q}$ can be represented by a unique function
$\epsilon_{p,q}:\mathcal{T}(p,q)\to\\{\pm 1\\}$; combining these functions, we
get a map $\epsilon:\mathcal{T}^{1}\to\\{\pm 1\\}$. By the descriptions of
$\partial_{1}\widehat{W}^{-}_{p}$ and
$\hat{\iota}^{-}_{p}:\partial_{1}\widehat{W}^{-}_{p}\to M$, and by the Stokes’
theorem for manifolds with corners, we have [9, Appendix by F. Laudenbach],
[32, Remark 1.9], [14, Theorem 3.6 and Proposition 5.3], [38, Section 6.5.3]

$\partial W^{-}_{p}=\sum_{q\in\mathcal{X}_{k-1},\
\gamma\in\mathcal{T}(p,q)}\epsilon(\gamma)\,W^{-}_{q}\;.$ (5.2)

Thus the currents $W^{-}_{p}$ ($p\in\mathcal{X}$) generate over $\mathbb{C}$ a
finite dimensional subcomplex $(C_{\bullet}(X,W^{-}),\partial)$ of the complex
$(\Omega(M)^{\prime},\partial)$ of currents on $M$, called the _Morse
complex_. The simpler notation
$\mathbf{C}_{\bullet}=\mathbf{C}_{\bullet}(X)=C_{\bullet}(X,W^{-})$ may be
also used. Moreover $\mathbf{C}_{\bullet}\hookrightarrow\Omega(M)^{\prime}$
induces an isomorphism444Actually, $H_{\bullet}(M,\mathbb{Z})$ is isomorphic
to the homology of the complex of free Abelian groups generated by the
currents $W^{-}_{p}$. $H_{\bullet}(\mathbf{C}_{\bullet},\partial)\cong
H_{\bullet}(M,\mathbb{C})$ [69, 65, 48] (see also [27, 62, 63], [33, Theorem
0.1], [9, Appendix by F. Laudenbach, Proposition 7], [38, Section 6.6.5]).

The _dual Morse complex_ is the dual $(C^{\bullet}(X,W^{-}),\mathbf{d})$ of
$(\mathbf{C}_{\bullet},\partial)$; namely,
$C^{k}(X,W^{-})=(\mathbf{C}_{k})^{*}\equiv\mathbb{C}^{\mathcal{X}_{k}}$
($k=0,\dots,n$). We will usually denote
$\mathbf{C}^{\bullet}=\mathbf{C}^{\bullet}(X)=C^{\bullet}(X,W^{-})$. Moreover
boldface notation will be used for elements of $\mathbf{C}^{\bullet}$ and
operators on $\mathbf{C}^{\bullet}$. Let $\mathbf{e}_{p}$ ($p\in\mathcal{X}$)
denote the elements of the canonical base of $\mathbf{C}^{\bullet}$,
determined by $\mathbf{e}_{p}(q)=\delta_{pq}$, using the Kronecker delta.
Then, for $q\in\mathcal{X}_{k-1}$,

$\mathbf{d}\mathbf{e}_{q}=\sum_{p\in\mathcal{X}_{k},\
\gamma\in\mathcal{T}(p,q)}\epsilon(\gamma)\,\mathbf{e}_{p}\;.$ (5.3)

####  5.2.2. The perturbed Morse complex

Take any $\eta\in Z^{1}(M,\mathbb{R})$ defining a class $\xi\in
H^{1}(M,\mathbb{R})$ (there is no need of any condition on $\eta$ or $g$ in
Sections 5.2.2, 5.2.3 and 5.2.4). For reasons of brevity, write
$\eta(\gamma)=\int_{\gamma}\eta$ for every $\gamma\in\mathcal{T}^{1}$.
According to [15, 16, 18], $(\mathbf{C}^{\bullet},\mathbf{d})$ has an analog
of the Witten’s perturbation,
$(\mathbf{C}^{\bullet},\mathbf{d}_{z}=\mathbf{d}_{z\eta})$ ($z\in\mathbb{C}$),
where, for $q\in\mathcal{X}_{k-1}$ ($k=1,\dots,n$),

$\mathbf{d}_{z}\mathbf{e}_{q}=\sum_{p\in\mathcal{X}_{k},\
\gamma\in\mathcal{T}(p,q)}\epsilon(\gamma)e^{z\eta(\gamma)}\mathbf{e}_{p}\;.$
(5.4)

If $\eta=dh$ for some $h\in C^{\infty}(M,\mathbb{R})$, then
$\mathbf{d}_{z}=e^{-zh}\mathbf{d}e^{zh}$ on $\mathbf{C}^{\bullet}$ because
$\eta(\gamma)=h(q)-h(p)$ for $p\in\mathcal{X}_{k}$, $q\in\mathcal{X}_{k-1}$
and $\gamma\in\mathcal{T}(p,q)$; here, $e^{\pm zh}$ also denotes the operator
of multiplication by the restriction of this function to $\mathcal{X}$. It
will be said that $(\mathbf{C}^{\bullet},\mathbf{d}_{z})$ ($z\in\mathbb{C}$)
is the _perturbed dual Morse complex_ defined by $X$ and $\eta$. A
perturbation $(\mathbf{C}_{\bullet},\partial^{z})$ is similarly defined,
adding the factor $e^{z\eta(\gamma)}$ to the terms of the right-hand side of
(5.2).

Since $W^{-}_{p}$ ($p\in\mathcal{X}_{k}$, $k=0,\dots,n$) is diffeomorphic to
$\mathbb{R}^{k}$, there is a unique $h_{\eta,p}^{-}\in
C^{\infty}(W^{-}_{p},\mathbb{R})$ such that $h_{\eta,p}^{-}(\hat{p}^{-})=0$
and $dh_{\eta,p}^{-}=(\iota_{p}^{-})^{*}\eta$. Indeed $h_{\eta,p}^{-}$ has a
smooth extension $\hat{h}_{\eta,p}^{-}$ to $\widehat{W}^{-}_{p}$ because
$\widehat{W}^{-}_{p}$ is contractile. By Proposition 5.1 ii, for all
$q\in\mathcal{X}_{k-1}$ and $\gamma\in\mathcal{T}(p,q)$, we have
$\hat{h}_{\eta,p}^{-}(\gamma,\hat{q}^{-})=\eta(\gamma)$, yielding

$\hat{h}_{\eta,q}^{-}\equiv\hat{h}_{\eta,p}^{-}-\eta(\gamma)\quad\text{on}\quad\widehat{W}^{-}_{q}\equiv\\{\gamma\\}\times\widehat{W}^{-}_{q}\subset\partial_{1}\widehat{W}^{-}_{p}\;.$

According to [15, Proposition 4], [16, Proposition 10], [18, Propositions 2.15
and 2.16 and Section 6.2], a sujective homomorphism of complexes,

$\Phi_{z}:(\Omega(M),d_{z})\to(\mathbf{C}^{\bullet},\mathbf{d}_{z})\;,$

is defined by

$\Phi_{z}(\omega)(p)=\int_{W^{-}_{p}}e^{zh_{\eta,p}^{-}}\omega=\int_{\widehat{W}^{-}_{p}}e^{z\hat{h}_{\eta,p}^{-}}(\hat{\iota}_{p}^{-})^{*}\omega\;.$

Moreover $\Phi_{z}$ is a quasi-isomorphism for all $z\in\mathbb{C}$ [9,
Proposition 1 in the Appendix by F. Laudenbach] (see also [9, Theorem 2.9],
[10, Theorem 1.6], [18, Proposition 2.17 and Section 6.2]). Then, by (3.14),
the same is true for

$\Phi_{z}:(E_{z,\text{\rm
sm}},d_{z})\to(\mathbf{C}^{\bullet},\mathbf{d}_{z})\;.$

Since direct adaptation of [16, Appendix A] shows that, for $k=0,\dots,n$,
$\dim H^{k}(\mathbf{C}^{\bullet},\mathbf{d}_{z})$ is independent of
$z\in\mathbb{C}$ with $|\mu|\gg 0$, we get (2.10).

####  5.2.3. Morse complex with coefficients in a flat vector bundle

With more generality [9, Section 1c)], for a flat vector bundle $F$, we may
consider $(C^{\bullet}(X,W^{-},F),\mathbf{d}^{F})$, where
$C^{k}(X,W^{-},F)=\bigoplus_{p\in\mathcal{X}_{k}}F_{p}$, and
$\mathbf{d}^{F}\mathbf{e}$ ($\mathbf{e}\in F_{q}$, $q\in\mathcal{X}_{k-1}$) is
defined like in the right-hand side of (5.3), replacing $\mathbf{e}_{p}$ with
the parallel transport of $\mathbf{e}$ along $\bar{\gamma}^{-1}$. This is the
dual of the complex $(C_{\bullet}(X,W^{-},F^{*}),\partial^{F^{*}})$, where
$C_{k}(X,W^{-},F^{*})=\bigoplus_{p\in\mathcal{X}_{k}}F_{p}^{*}$, and
$\partial^{F}f$ ($f\in F_{p}^{*}$, $p\in\mathcal{X}_{k}$) is defined like in
the right-hand side of (5.2), replacing $W^{-}_{q}$ with the parallel
transport of $f$ along $\bar{\gamma}$. A homomorphism

$\Phi^{F}=\Phi^{X,F}:(\Omega(M,F),d)\to\big{(}C^{\bullet}(X,W^{-},F),\mathbf{d}^{F}\big{)}$

can be defined like $\Phi_{z}$, using the isomorphism

$\Omega^{\bullet}\big{(}\widehat{W}^{-}_{p},(\hat{\iota}^{-}_{p})^{*}F\big{)}\cong\Omega^{\bullet}\big{(}\widehat{W}^{-}_{p}\big{)}\otimes
F_{p}$

given by the parallel transport of $(\hat{\iota}^{-}_{p})^{*}F$. With this
generality, $\Phi^{F}$ is also induces a quasi-isomorphism [9, Theorem 2.9].
If $F=\mathcal{L}^{z}$ (Section 2.1.4), then

$\big{(}C^{\bullet}(X,W^{-},\mathcal{L}^{z}),\mathbf{d}^{\mathcal{L}^{z}}\big{)}\equiv(\mathbf{C}^{\bullet},\mathbf{d}_{z})\;,\quad\Phi^{\mathcal{L}^{z}}\equiv\Phi_{z}\;.$

####  5.2.4. Hodge theory of the Morse complex

Consider the Hermitian scalar product on $\mathbf{C}^{\bullet}$ so that the
canonical base $\mathbf{e}_{p}$ ($p\in\mathcal{X}$) is orthonormal. All
operators induced by $\mathbf{d}_{z}$ and this Hermitian structure are called
_perturbed Morse operators_. For instance, besides $\mathbf{d}_{z}$, we have
the perturbed Morse operators

$\displaystyle\boldsymbol{\delta}_{z}=\mathbf{d}_{z}^{*}\;,\quad\mathbf{D}_{z}=\mathbf{d}_{z}+\boldsymbol{\delta}_{z}\;,\quad\boldsymbol{\Delta}_{z}=\mathbf{D}_{z}^{2}=\mathbf{d}_{z}\boldsymbol{\delta}_{z}+\boldsymbol{\delta}_{z}\mathbf{d}_{z}\;.$

In particular, it will be said that $\boldsymbol{\Delta}_{z}$ is the
_perturbed Morse Laplacian_ , and its eigenvalues will be called _perturbed
Morse eigenvalues_. If $z=0$, we omit the subscript “$z$” and the word
“perturbed”. From (5.4), we easily get

$\boldsymbol{\delta}_{z}\mathbf{e}_{p}=\sum_{q\in\mathcal{X}_{k-1},\
\gamma\in\mathcal{T}(p,q)}e^{\bar{z}\eta(\gamma)}\epsilon(\gamma)\,\mathbf{e}_{q}\;,$
(5.5)

for $p\in\mathcal{X}_{k}$. We also have

$\displaystyle\ker\boldsymbol{\Delta}_{z}$
$\displaystyle=\ker\mathbf{D}_{z}=\ker\mathbf{d}_{z}\cap\ker\boldsymbol{\delta}_{z}\;,$
$\displaystyle\operatorname{im}\boldsymbol{\Delta}_{z}$
$\displaystyle=\operatorname{im}\mathbf{D}_{z}=\operatorname{im}\mathbf{d}_{z}\oplus\operatorname{im}\boldsymbol{\delta}_{z}\;,$
$\displaystyle\mathbf{C}^{\bullet}$
$\displaystyle=\ker\boldsymbol{\Delta}_{z}\oplus\operatorname{im}\mathbf{d}_{z}\oplus\operatorname{im}\boldsymbol{\delta}_{z}\;.$

The orthogonal projections of $\mathbf{C}^{\bullet}$ to
$\ker\boldsymbol{\Delta}_{z}$, $\operatorname{im}\mathbf{d}_{z}$ and
$\operatorname{im}\boldsymbol{\delta}_{z}$ are denoted by
$\boldsymbol{\Pi}_{z}=\boldsymbol{\Pi}^{0}_{z}$, $\boldsymbol{\Pi}_{z}^{1}$
and $\boldsymbol{\Pi}_{z}^{2}$, respectively. The compositions
$\mathbf{d}_{z}^{-1}\boldsymbol{\Pi}^{1}$,
$\boldsymbol{\delta}_{z}^{-1}\boldsymbol{\Pi}^{2}$ and
$\mathbf{D}_{z}^{-1}\boldsymbol{\Pi}^{\perp}$ are defined like in Section
2.1.2, and there is an obvious version of the commutative diagram (2.8).

###  5.3. The small complex vs the Morse complex

Our main objects of interest are the form $\eta\in Z^{1}(M;\mathbb{R})$ and
the Riemannian metric $g$; $X$ plays an auxiliary role. Unless otherwise
indicated, assume from now on that $\eta$ is Lyapunov for $X$, and $\eta$ and
$g$ are in standard form with respect to $X$; i.e., we assume c besides a and
b. Since every $\xi\in H^{1}(M,\mathbb{R})$ is Lyapunov for $X$ by b, we can
choose some $\eta\in\xi$ and $g$ satisfying a and c.

For every $p\in\mathcal{X}$, consider the functions $h_{\eta,p}$, $h_{X,p}$,
$h_{\eta,p}^{-}$ and $\hat{h}_{\eta,p}^{-}$ defined in Sections 3.1, 5.1.1 and
5.2.2. Since $\eta$ and $g$ are in standard form with respect to $X$, we have
$h_{\eta,p}=h_{X,p}$ on $U_{p}$, and

$h_{\eta,p}^{-}=h_{\eta,p}=-\frac{1}{2}|x_{p}^{-}|^{2}$ (5.6)

on $U_{p}^{-}$. From now on, the simpler notation $h_{p}=h_{\eta,p}=h_{X,p}$,
$h_{p}^{-}=h_{\eta,p}^{-}$ and $\hat{h}_{p}^{-}=\hat{h}_{\eta,p}^{-}$ will be
used. Since $\eta$ is Lyapunov for $X$,

$h_{p}^{-}<0\quad\text{on}\quad W^{-}_{p}\setminus\\{p\\}\;.$ (5.7)

Consider the notation of Section 5.2.2. Let $J_{z}:\mathbf{C}^{\bullet}\to
E_{z}$ be the $\mathbb{C}$-linear isometry given by
$J_{z}(\mathbf{e}_{p})=e_{p,z}$, and let $\Psi_{z}=P_{z,\text{\rm
sm}}J_{z}:\mathbf{C}^{\bullet}\to E_{z,\text{\rm sm}}$, which is an
isomorphism for $\mu\gg 0$ (Corollary 3.9). By Proposition 3.7,

$\|\Psi_{z}\mathbf{e}\|=\big{(}1+O\big{(}e^{-c\mu}\big{)}\big{)}\|\mathbf{e}\|\quad(\mu\to+\infty)$

for all $\mathbf{e}\in\mathbf{C}^{\bullet}$. Using polarization (see e.g. [35,
Section I.6.2]) and conjugation, this means that, as $\mu\to+\infty$,