(200) $p(\frac{\partial}{\partial h})\left({vol}\
{P(h)}\right)_{|_{h=0}}=\chi(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}),$

with ${vol}\ {P(h)}=\int_{P(h)}dm$ the volume of $P(h)$, and the Lebesgue
measure normalized so that the unit cube in $M\subset M_{\mathbb{R}}$ has
volume $1$.

######  Example 5.20.

The classical volume formula [12][Thm.2.15] corresponds to
${\mathcal{F}}={\mathcal{O}}_{X}$ for $p$ given by the Todd operator
$Todd(\frac{\partial}{\partial h})$ of (203) below, with
$\chi(X,{\mathcal{O}}_{X}(D))=|P\cap M|$. ∎

######  Corollary 5.21.

Let $X=X_{P}$ be the projective simplicial toric variety associated to a full-
dimensional simple lattice polytope $P\subset M_{\mathbb{R}}$. Let
$\Sigma:=\Sigma_{P}$ be the inner normal fan of $P$, and $D:=D_{P}$ the ample
Cartier divisor associated to $P$. Let $[{\mathcal{F}}]\in
K^{\mathbb{T}}_{0}(X)$ be fixed, and choose a formal power series
$p(x_{\rho})\in{\mathbb{Q}}[[x_{\rho}\mid\rho\in\Sigma(1)]]$ so that
$p(F_{\rho})={\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}])\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$.
Then for a polynomial function $f$ on $M_{\mathbb{R}}$, we have:

(201) $p(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
f(m)\>.$

###### Proof.

This follows from Corollary 5.14 by applying the operator
$\left(f(\frac{\partial}{\partial z})\right)_{|_{z=0}}$ to the last term of
formula (199), seen as a formal power series in $z$. ∎

Let us finish this section with a local counterpart of Theorem 5.18. Using
Remark 5.10, Corollary 5.16 and Proposition 4.13, we get:

######  Proposition 5.22.

Let $X=X_{P}$ be the projective simplicial toric variety associated to a full-
dimensional simple lattice polytope $P\subset
M_{\mathbb{R}}\simeq{\mathbb{R}}^{n}$. Let $\Sigma:=\Sigma_{P}$ be the inner
normal fan of $P$, and $D:=D_{P}$ the ample Cartier divisor associated to $P$.
Let $[{\mathcal{F}}]\in K^{\mathbb{T}}_{0}(X)$ be fixed, and choose a
convergent power series
$p(x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}$ so that
$p(F_{\rho})={\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}])\in\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}$.
Fix a vertex $v\in P$ with tangent cone $Tan(P,v)$, and let
$\sigma\in\Sigma(n)$ be the corresponding cone. Then, for $z$ small enough
with $-z\in Int(\sigma)$, we have the following formula:

(202) $\begin{split}p(\frac{\partial}{\partial
h})\left(\int_{Tan(P,v)(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}&=\langle e^{v}\cdot{\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])_{x_{\sigma}},z\rangle\\\ &=\langle
e^{v}\cdot{\rm
ch}^{\mathbb{T}}((\mathbb{S}\circ\chi^{\mathbb{T}}_{\sigma})({\mathcal{F}})),z\rangle\>.\end{split}$

###  5.1. Examples of Euler-Maclaurin formulae

In this section, we explain how various special cases of Corollary 5.21 and
Proposition 5.22 yield old and new Euler-Maclaurin type formulae.

In the global context of Corollary 5.21, let $X=X_{P}$ be the projective
simplicial toric variety associated to a full-dimensional simple lattice
polytope $P\subset M_{\mathbb{R}}$. Let $\Sigma:=\Sigma_{P}$ be the inner
normal fan of $P$, and $D:=D_{P}$ the ample Cartier divisor associated to $P$.
We fix a polynomial $g$ on $M_{\mathbb{R}}$, and let $f(m)=g(m)\cdot
e^{\langle m,z\rangle}$ be a quasi-polynomial, with $z\in N_{\mathbb{C}}$
small enough. In the context of a tangent cone at a vertex $v$ of $P$ as in
Proposition 5.22, we only use an exponential function $f(m)=e^{\langle
m,z\rangle}$ with $z\in N_{\mathbb{C}}$ small enough and $-z\in Int(\sigma)$.
In this local case, $p(x)$ is restricted to the variables $F_{\rho}$,
$\rho\in\sigma(1)$ (as in Remark 5.10), with $\sigma\in\Sigma(n)$ the cone
corresponding to $v$. In the concrete formulas below, this amounts to using
the cone $\sigma$ as a fan instead of $\Sigma$, and the finite group
$G_{\sigma}$ instead of $G_{\Sigma}$.

######  Example 5.23.

The first case we consider is the classical one of the Euler-Maclaurin formula
for the polytope $P$. Here we choose ${\mathcal{F}}={\mathcal{O}}_{X}$ and the
infinite order differential operator

(203) $Todd(\frac{\partial}{\partial h}):=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{\frac{\partial}{\partial
h_{\rho}}}{1-a_{\rho}(g)\cdot e^{-\frac{\partial}{\partial
h_{\rho}}}}\in\mbox{$\mathbb{Q}$}\\{\frac{\partial}{\partial
h_{\rho}}\;|\;\rho\in\Sigma(1)\\}\>,$

as dictated by formula (130) for ${\rm td}_{*}^{\mathbb{T}}(X)$. Then the
equivariant Euler characteristic formula (75) yields the Euler-Maclaurin
formula of Brion-Vergne [13][Thm.4.5], extended here to a quasi-polynomial
$f$:

(204) $Todd(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in P\cap M}f(m)\>.$

As explain before, the left-hand side of (204) only depends of the class of
$Todd(\frac{\partial}{\partial h})$ in the analytic Stanley-Reisner ring of
$X$. For example, another such representative corresponds to the power series
fitting with (the equivariant version of) Theorem 2.6, expressing the Todd
operator in terms of suitable $L$-class versions. If $P$ is a Delzant
polytope, one recovers formula (184) of Khovanskii-Pukhlikov, fitting also
with the equivariant Todd class of smooth projective toric varieties (e.g.,
obtained by setting $y=0$ in Corollary 3.23).

In the local case of the tangent cone $Tan(P,v)$ of $P$ at a vertex $v$, using
(191) we get (see also [38][Thm.1], [34][eqn.(B.1)]):

(205) $\begin{split}Todd(\frac{\partial}{\partial
h})\left(\int_{Tan(P,v)(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}=e^{\langle v,z\rangle}\cdot\sum_{m\in Tan(P,v)\cap
M}e^{\langle m,z\rangle}\>.\end{split}$

∎

######  Example 5.24.

An Euler-Maclaurin formula for the interior of a simple lattice polytope $P$
can be obtained similarly by using the dual Todd operator

(206) $Todd^{\vee}(\frac{\partial}{\partial h}):=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{a_{\rho}(g)\cdot\frac{\partial}{\partial
h_{\rho}}\cdot e^{-\frac{\partial}{\partial h_{\rho}}}}{1-a_{\rho}(g)\cdot
e^{-\frac{\partial}{\partial
h_{\rho}}}}\in\mbox{$\mathbb{Q}$}\\{\frac{\partial}{\partial
h_{\rho}}\;|\;\rho\in\Sigma(1)\\}\>,$

corresponding to the sheaf ${\mathcal{F}}=\omega_{X}$ and its dual Todd class
${\rm td}_{*}^{\mathbb{T}}([\omega_{X}]_{\mathbb{T}})$ of formula (131). In
particular, by the equivariant Euler characteristic formula (76) one gets:

(207) $Todd^{\vee}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in{\rm Int}(P)\cap M}f(m)\>.$

In the local case of the tangent cone $Tan(P,v)$ of $P$ at a vertex $v$, using
(192) we get:

(208) $\begin{split}Todd^{\vee}(\frac{\partial}{\partial
h})\left(\int_{Tan(P,v)(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}=e^{\langle v,z\rangle}\cdot\sum_{m\in Int(Tan(P,v))\cap
M}e^{\langle m,z\rangle}\>.\end{split}$

∎

More generally, one can prove an Euler-Maclaurin formula for a polytope with
some facets removed, see also [34][Prop.7.2] for the case of Delzant
polytopes. Let $P$ be a full-dimensional polytope with $r$ facets
$F_{1},\ldots,F_{r}$. For a subset $K\subseteq\\{1,\ldots,r\\}$, let $P^{K}$
be the set obtained from $P$ by removing the facets $F_{i}$, $i\in K$. For
example, $P^{\emptyset}=P$ and $P^{\\{1,\ldots,r\\}}={\rm Int}(P)$. Let
$\Sigma(1)=\\{\rho_{1},\ldots,\rho_{r}\\}$ be the rays of the inner normal fan
of $P$, and denote by $h=(h_{1},\ldots,h_{r})$ a vector of real numbers
indexed by the facets of $P$ (i.e., $h_{i}:=h_{\rho_{i}}$). Consider the
following operator in ${\mathbb{Q}}\\{\frac{\partial}{\partial h_{i}}\mid
i=1,\ldots,r\\}$:

(209) $Todd^{K}(\frac{\partial}{\partial h}):=\sum_{g\in
G_{\Sigma}}\prod_{i\in K}\frac{a_{\rho_{i}}(g)\cdot\frac{\partial}{\partial
h_{i}}\cdot e^{-\frac{\partial}{\partial h_{i}}}}{1-a_{\rho_{i}}(g)\cdot
e^{-\frac{\partial}{\partial h_{i}}}}\cdot\prod_{i\notin
K}\frac{\frac{\partial}{\partial h_{i}}}{1-a_{\rho_{i}}(g)\cdot
e^{-\frac{\partial}{\partial h_{i}}}},$

corresponding to formula (139) for the equivariant Todd class of
$[{\mathcal{F}}]=mC_{0}^{\mathbb{T}}([U\hookrightarrow X])$, with
$U=X\setminus D_{K}$ the open complement of the divisor $D_{K}=\bigcup_{i\in
K}D_{i}$ and $D_{i}:=D_{\rho_{i}}$. Moreover, by (140) and Corollary 3.15 for
$y=0$, applied to $X$ and $D_{K}$, one gets a corresponding equivariant
Hirzebruch-Riemann-Roch formula for ${\rm
td}_{*}^{\mathbb{T}}({\mathcal{O}}_{X}(D)\otimes
mC_{0}^{\mathbb{T}}([U\hookrightarrow X]))$, namely

$\int_{X}{\rm td}_{*}^{\mathbb{T}}({\mathcal{O}}_{X}(D)\otimes
mC_{0}^{\mathbb{T}}([U\hookrightarrow X]))=\sum_{m\in P^{K}\cap M}e^{s(m)}.$

For its local counterpart, we use (172) for $y=0$. With the above notations,
together with (192), this then gives the following.

######  Theorem 5.25.

Let $P$ be a full-dimensional simple lattice polytope in $M_{\mathbb{R}}$, and
let $f$ be a polynomial function on $M_{\mathbb{R}}$. Then:

(210) $Todd^{K}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in P^{K}\cap M}f(m).$

In the local case of the tangent cone $Tan(P,v)$ of $P$ at a vertex $v$, we
get:

(211) $\begin{split}Todd^{K}(\frac{\partial}{\partial
h})\left(\int_{Tan(P,v)(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}=e^{\langle v,z\rangle}\cdot\sum_{m\in Tan(P,v)^{K}\cap
M}e^{\langle m,z\rangle}\>.\end{split}$

We next explain an Euler-Maclaurin formula for the interior of a face of a
simple lattice polytope. Let $P$ be a full-dimensional simple lattice polytope
in $M_{\mathbb{R}}$, and fix a face $E$ of $P$. Let $\sigma:=\sigma_{E}$ be
the corresponding cone in the inner normal fan $\Sigma=\Sigma_{P}$ of $P$,
with $V_{\sigma}=V_{\sigma_{E}}=X_{E}$ the closure of the orbit of $\sigma$ in
$X=X_{P}$. Denote by $i_{E}=i_{\sigma}:V_{\sigma}\hookrightarrow X$ the closed
inclusion map. Then $V_{\sigma}$ is a simplicial toric variety whose fan is
$Star(\sigma)$, as defined in (28), which is built from cones $\tau\in\Sigma$
that have $\sigma$ as a face. Recall that ${\mathbb{T}}=T_{N}$ acts on
$V_{\sigma}$ via the morphism $T_{N}\to T_{N(\sigma)}$. Then we get, as in
Corollary 3.25 and with the above notations, the following formula:

(212) ${\rm
td}_{*}^{\mathbb{T}}([\omega_{V_{\sigma}}]_{\mathbb{T}})=i_{\sigma}^{*}\left(\sum_{g\in
G_{Star(\sigma)}}\prod_{\rho\in Star(\sigma)(1)}\frac{a_{\rho}(g)\cdot
F_{\rho}\cdot e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\right).$

Together with (62) and the projection formula, this gives the equivariant Todd
class of $[{\mathcal{F}}]=[(i_{\sigma})_{*}\omega_{V_{\sigma}}]$:

(213) ${\rm
td}_{*}^{\mathbb{T}}([(i_{\sigma})_{*}\omega_{V_{\sigma}}])=\sum_{g\in
G_{Star(\sigma)}}{\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}F_{\rho}\cdot\prod_{\rho\in
Star(\sigma)(1)}\frac{a_{\rho}(g)\cdot F_{\rho}\cdot
e^{-F_{\rho}}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}.$

Using (120), this yields the following result (where we use the above
notations):

######  Theorem 5.26.

Let $P$ be a full-dimensional simple lattice polytope in $M_{\mathbb{R}}$, and
let $f$ be a polynomial function on $M_{\mathbb{R}}$. Then, for a fixed face
$E$ of $P$, with corresponding cone $\sigma$ in the inner normal fan $\Sigma$
of $P$, one has:

(214) $Todd^{\vee}_{E}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in{\rm Relint}(E)\cap M}f(m),$

where

(215) $Todd^{\vee}_{E}(\frac{\partial}{\partial h}):=\sum_{g\in
G_{Star(\sigma)}}{\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}\frac{\partial}{\partial
h_{\rho}}\cdot\prod_{\rho\in
Star(\sigma)(1)}\frac{a_{\rho}(g)\cdot\frac{\partial}{\partial h_{\rho}}\cdot
e^{-\frac{\partial}{\partial h_{\rho}}}}{1-a_{\rho}(g)\cdot
e^{-\frac{\partial}{\partial h_{\rho}}}}.$

We next discuss a similar Euler-Maclaurin formula for a face of a simple
lattice polytope. With the same notations, we have by (130) the following
formula:

(216) ${\rm
td}_{*}^{\mathbb{T}}([{\mathcal{O}}_{V_{\sigma}}]_{\mathbb{T}})=i_{\sigma}^{*}\left(\sum_{g\in
G_{Star(\sigma)}}\prod_{\rho\in
Star(\sigma)(1)}\frac{F_{\rho}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\right).$

As before, the projection formula yields the equivariant Todd class of
$[{\mathcal{F}}]=[(i_{\sigma})_{*}{\mathcal{O}}_{V_{\sigma}}]$:

(217) ${\rm
td}_{*}^{\mathbb{T}}([(i_{\sigma})_{*}{\mathcal{O}}_{V_{\sigma}}])=\sum_{g\in
G_{Star(\sigma)}}{\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}F_{\rho}\cdot\prod_{\rho\in
Star(\sigma)(1)}\frac{F_{\rho}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}.$

Using (121), this yields the following result:

######  Theorem 5.27.

Let $P$ be a full-dimensional simple lattice polytope in $M_{\mathbb{R}}$, and
let $f$ be a polynomial function on $M_{\mathbb{R}}$. Then, for a fixed face
$E$ of $P$, with corresponding cone $\sigma$ in the inner normal fan $\Sigma$
of $P$, one has:

(218) $Todd_{E}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in E\cap M}f(m),$

where

(219) $Todd_{E}(\frac{\partial}{\partial h}):=\sum_{g\in G_{Star(\sigma)}}{\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}\frac{\partial}{\partial
h_{\rho}}\cdot\prod_{\rho\in Star(\sigma)(1)}\frac{\frac{\partial}{\partial
h_{\rho}}}{1-a_{\rho}(g)\cdot e^{-\frac{\partial}{\partial h_{\rho}}}}.$

Other interesting coherent sheaves to consider are the Zariski sheaves
$\widehat{\Omega}^{p}_{X}$ of $p$-forms on the toric variety $X$. In the next
section, these will be considered all at once via the formal sum

$\bigoplus_{p}[\widehat{\Omega}^{p}_{X}]_{\mathbb{T}}\cdot y^{p}\in
K^{\mathbb{T}}_{0}(X)[y],$

and similarly for suitable motivic Chern classes $mC^{\mathbb{T}}_{y}$ of
${\mathbb{T}}$-invariant constructible subsets of $X$.

We leave it to the reader to specialize the generalized volume formula (200)
to all situations discussed in this section.

Another way to obtain examples of explicit Euler-Maclaurin formulae is by
twisting the coherent sheaf ${\mathcal{F}}$ by
${\mathcal{O}}_{X}(D^{\prime}-D)$, for $D=D_{P}$ the original ample divisor
associated to the full-dimensional lattice polytope, and $D^{\prime}$ any
${\mathbb{T}}$-invariant Cartier divisor on $X$. By the multiplicativity of
the equivariant Todd class transformation for the coherent sheaf
${\mathcal{F}}^{\prime}={\mathcal{O}}_{X}(D^{\prime}-D)\otimes{\mathcal{F}}$,
we have

${\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}^{\prime}])=e^{[D^{\prime}-D]_{\mathbb{T}}}\cdot{\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}]).$

So, if $p(x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}$ is a
convergent power series with
$p(F_{\rho})=td_{*}^{\mathbb{T}}([{\mathcal{F}}])$, then
$p^{\prime}(x_{\rho}):=e^{\sum_{\rho\in\Sigma(1)}d_{\rho}x_{\rho}}\cdot
p(x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}$ is a convergent
power series with
$p^{\prime}(F_{\rho})=td_{*}^{\mathbb{T}}([{\mathcal{F}}^{\prime}])$, where
$D^{\prime}-D=\sum_{\rho\in\Sigma(1)}d_{\rho}D_{\rho}$ as a
${\mathbb{T}}$-invariant Cartier divisor.

######  Example 5.28.

As a last concrete example of this section, assume $D^{\prime}$ is a globally
generated ${\mathbb{T}}$-invariant Cartier divisor on $X$, with associated
(not necessarily full-dimensional) lattice polytope $P_{D^{\prime}}\subset
M_{\mathbb{R}}$. Consider the infinite order differential operator

(220) $Todd^{\prime}(\frac{\partial}{\partial
h}):=e^{\sum_{\rho\in\Sigma(1)}d_{\rho}\cdot\frac{\partial}{\partial
h_{\rho}}}\cdot Todd(\frac{\partial}{\partial h})$

with $Todd(\frac{\partial}{\partial h})$ as in (203) and $d_{\rho}$’s as
above. Then the equivariant Euler characteristic formula (79) yields the
following new Euler-Maclaurin formula for a quasi-polynomial $f$.

(221) $Todd^{\prime}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in P_{D^{\prime}}\cap M}f(m)\>.$

Note that $P_{D^{\prime}}$ is an ${\mathbb{N}}$-Minkowski summand (in the
sense of [16][Def.6.2.11]) of the original polytope $P$, and any such
${\mathbb{N}}$-Minkowski summand comes from a globally generated Cartier
divisor $D^{\prime}$, see [16][Cor.6.2.15]. ∎

##  6\. Weighted Euler-Maclaurin formulae

Let $X=X_{P}$ be the projective simplicial toric variety associated to a full-
dimensional simple lattice polytope $P\subset
M_{\mathbb{R}}\simeq{\mathbb{R}}^{n}$. Let $\Sigma:=\Sigma_{P}$ be the inner
normal fan of $P$, and $D:=D_{P}$ the ample Cartier divisor associated to $P$.

Let a convergent power series
$p(x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}$ be given.
Additionally, we may start with a polynomial in $y$ with coefficients
consisting of such power series, as we will need in the applications of this
section. Define a corresponding renormalized series

$p_{y}(x_{\rho}):=\frac{p\left((1+y)x_{\rho}\right)}{(1+y)^{n}}\in{\mathbb{Q}}[y,(1+y)^{-1}][[x_{\rho}\mid\rho\in\Sigma(1)]].$

To treat it as a convergent power series, one needs to assume that
$y\in{\mathbb{R}}\setminus\\{1\\}$ is fixed or it belongs to a compact subset.

Let $p_{y}(\frac{\partial}{\partial h})$ be the corresponding parametrized
infinite order differential operator obtained from $p_{y}(x_{\rho})$ by
substituting $x_{\rho}\mapsto\frac{\partial}{\partial h_{\rho}}$, for all
$\rho\in\Sigma(1)$. Then formula (196) translates into the following:

(222) $p_{y}(\frac{\partial}{\partial h})\left(\int_{P_{y}(h)}e^{\langle
m,z\rangle}\
dm\right)_{|_{h=0}}=\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle(1+y)\cdot
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P})),z\rangle}}{(1+y)^{n}\cdot\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\langle p((1+y)\cdot
i_{\sigma}^{*}F_{\rho}),z\rangle\>.$

Let $[{\mathcal{F}}]\in K^{\mathbb{T}}_{0}(X)$ be fixed, and choose a
convergent power series
$p(x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}$ so that
$p(F_{\rho})={\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}])\in\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}$.
Applying the proof of Theorem 5.18 to formula (222), one then gets

(223) $p_{y}(\frac{\partial}{\partial h})\left(\int_{P_{y}(h)}e^{\langle
m,z\rangle}\ dm\right)_{|_{h=0}}=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
e^{\langle(1+y)m,z\rangle}\>.$

Here the use of the renormalized power series $p_{y}$ and polytope $P_{y}(h)$
correspond to multiplying each degree $2k$ equivariant cohomology class by
$(1+y)^{k}$. (This is just the cohomological Adams operation $\Psi^{(1+y)}$.)
In particular, via the identification $s:M\simeq
H^{2}_{\mathbb{T}}(pt;{\mathbb{Z}})$, $m\in M$ gets multiplied by $1+y$.

For any polynomial $f$ defined on $M_{\mathbb{R}}$, by applying the operator
$\left(f(\frac{\partial}{\partial z})\right)_{|_{z=0}}$ to formula (223), we
get the following parametrized Euler-Maclaurin formula:

(224) $p_{y}(\frac{\partial}{\partial h})\left(\int_{P_{y}(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
f\left((1+y)m\right)\>.$

Moreover, if $f$ is homogeneous of degree $\deg(f)$, then
$f\left((1+y)m\right)=(1+y)^{\deg(f)}\cdot f(m).$

######  Remark 6.1.

In the local case of Euler-Maclaurin formulae for a tangent cone $Tan(P,v)$,
we only work with the unnormalized equivariant Hirzebruch classes
$T^{\mathbb{T}}_{y*}$ and the dilatation $Tan(P,v)(h)$ of the tangent cone. In
this context, a renormalization is not needed.

###  6.1. Weighted Euler-Maclaurin formulae

In this section, we apply the above renormalization not just to classes of
coherent sheaves, but directly to examples of the type

$\bigoplus_{p}[\widehat{\Omega}^{p}_{X}]_{\mathbb{T}}\cdot y^{p}\in
K^{\mathbb{T}}_{0}(X)[y],$

as well as for suitable motivic Chern classes $mC^{\mathbb{T}}_{y}$ of
${\mathbb{T}}$-invariant constructible subsets of $X$.

Recall from Section 3.3 that the equivariant Hirzebruch classes
${T}^{\mathbb{T}}_{y*}(X)\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})[y]$,
resp.,
$\widehat{T}^{\mathbb{T}}_{y*}(X)\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}}[y])$
of $X$ are computed as:

(225) ${T}^{\mathbb{T}}_{y*}(X)=(1+y)^{n-r}\cdot\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot\big{(}1+y\cdot
a_{\rho}(g)\cdot e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\>,$
(226) $\widehat{T}^{\mathbb{T}}_{y*}(X)=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{F_{\rho}\cdot\big{(}1+y\cdot
a_{\rho}(g)\cdot e^{-F_{\rho}(1+y)}\big{)}}{1-a_{\rho}(g)\cdot
e^{-F_{\rho}(1+y)}}\>,$

with $r=|\Sigma(1)|$, and $F_{\rho}=[D_{\rho}]_{\mathbb{T}}$ denoting the
equivariant fundamental class of the ${\mathbb{T}}$-invariant divisor
$D_{\rho}$ corresponding to the ray $\rho\in\Sigma(1)$.

The weighted lattice point counting of Theorem 2.3 (and Remark 2.4), coupled
with the above expressions for the equivariant Hirzebruch classes, suggest
that a weighted Euler-Maclaurin-type formula can be computed by using the
Hirzebruch (or, parametrized Todd) differential operators defined by:

${T}_{y}(\frac{\partial}{\partial h}):=(1+y)^{n-r}\cdot\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{\frac{\partial}{\partial
h_{\rho}}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot e^{-\frac{\partial}{\partial
h_{\rho}}}\big{)}}{1-a_{\rho}(g)\cdot e^{-\frac{\partial}{\partial
h_{\rho}}}}$

and

$\widehat{T}_{y}(\frac{\partial}{\partial h}):=\sum_{g\in
G_{\Sigma}}\prod_{\rho\in\Sigma(1)}\frac{\frac{\partial}{\partial
h_{\rho}}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot e^{-\frac{\partial}{\partial
h_{\rho}}(1+y)}\big{)}}{1-a_{\rho}(g)\cdot e^{-\frac{\partial}{\partial
h_{\rho}}(1+y)}}.$

The passage from ${T}_{y}(\frac{\partial}{\partial h})$ to
$\widehat{T}_{y}(\frac{\partial}{\partial h})$ is just a special case of our
general procedure of moving from $p(\frac{\partial}{\partial h})$ to
$p_{y}(\frac{\partial}{\partial h})$. This therefore fits with the
renormalization of Hirzebruch classes, from ${T}^{\mathbb{T}}_{y*}(X)$ to
$\widehat{T}^{\mathbb{T}}_{y*}(X)$.

Weighted Euler-Maclaurin formulae were considered from a combinatorial point
of view in [30][Thm.1.1 and Rem.1.2] for ${T}_{y}(\frac{\partial}{\partial
h})$, and more recently in [5][Thm.1.5] for
$\widehat{T}^{\mathbb{T}}_{y*}(X)$. We reprove here the following result (see
[30][Thm.1.1 and Rem.1.2] for ${T}_{y}(\frac{\partial}{\partial h})$, and
[5][Thm.4.6] for $\widehat{T}_{y}(\frac{\partial}{\partial h})$), from the
point of view of generalized Hirzebruch-Riemann-Roch formula:

######  Theorem 6.2.

For $z\in N_{\mathbb{C}}$ small enough, one has in the above notations:

(227) ${T}_{y}(\frac{\partial}{\partial h})\left(\int_{P(h)}e^{\langle
m,z\rangle}dm\right)_{|_{h=0}}=\sum_{E\preceq P}(1+y)^{\dim(E)}\sum_{m\in{\rm
Relint}(E)\cap M}e^{\langle m,z\rangle}.$ (228)
$\widehat{T}_{y}(\frac{\partial}{\partial h})\left(\int_{P_{y}(h)}e^{\langle
m,z\rangle}dm\right)_{|_{h=0}}=\sum_{E\preceq P}(1+y)^{\dim(E)}\sum_{m\in{\rm
Relint}(E)\cap M}e^{\langle(1+y)m,z\rangle}.$

In the corresponding local case of the tangent cone $Tan(P,v)$ of $P$ at a
vertex $v$, for $z\in N_{\mathbb{C}}$ small enough with $-z\in Int(\sigma)$,we
get

(229) $\begin{split}T_{y}(\frac{\partial}{\partial
h})\left(\int_{Tan(P,v)(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}=\sum_{E\preceq Tan(P,v)}(1+y)^{\dim(E)}\cdot e^{\langle
v,z\rangle}\cdot\sum_{m\in{\rm Relint}(E)\cap M}e^{\langle
m,z\rangle}\>.\end{split}$

###### Proof.

To deduce formula (227), instead of $[{\mathcal{F}}]\in K^{\mathbb{T}}_{0}(X)$
one considers the motivic Chern class $mC^{\mathbb{T}}_{y}(X)\in
K^{\mathbb{T}}_{0}(X)[y]$, with $T_{y*}(X):={\rm
td}_{*}^{\mathbb{T}}(mC^{\mathbb{T}}_{y}(X))$. We work with the power series
$p(x_{\rho}):=T_{y}(x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}[y]$
fitting with the differential operator ${T}_{y}(\frac{\partial}{\partial h})$.
Formula (227) follows now from (199) (linearly extended in $y$), using the
equivariant Hirzebruch-Riemann-Roch formula of Corollary 3.15, applied to the
case $P^{\prime}=P$. Formula (228) is then obtained via renormalization using
(223).

For the local result, we use (166) together with (192). ∎

Using (224), one then obtains the following result (see also [30][Thm.1.1 and
Rem.1.2] and [5][Thm.1.5]):

######  Corollary 6.3.

For any polynomial function $f$ on $M_{\mathbb{R}}$, one has:

(230) ${T}_{y}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{E\preceq P}(1+y)^{\dim(E)}\sum_{m\in{\rm
Relint}(E)\cap M}f(m).$ (231) $\widehat{T}_{y}(\frac{\partial}{\partial
h})\left(\int_{P_{y}(h)}f(m)\ dm\right)_{|_{h=0}}=\sum_{E\preceq
P}(1+y)^{\dim(E)}\sum_{m\in{\rm Relint}(E)\cap M}f((1+y)m).$

Note that the Brion-Verne Euler-Maclaurin formula for the simple lattice
polytope $P$ is obtained from either (230) or (231) by specializing to $y=0$.
Moreover, for $y=1$, one gets Euler-Maclaurin formulas corresponding to
operators related to suitable $L$-classes, i.e.,
$\widehat{T}_{1*}(X)=L_{*}(X)$ the Thom-Milnor $L$-class of $X$ (see [36]).
Taking the top degree in $y$ on both sides of (230) recovers formula (207).

Next we prove a weighted Euler-Maclaurin formula for a polytope with some
facets removed. Here and below, for simplicity, we only indicate such a
formula for a polynomial $f$, while we leave it to the reader to formulate the
corresponding exponential formula. Let $P$ be a full-dimensional polytope with
$r$ facets $F_{1},\ldots,F_{r}$. For a subset $K\subseteq\\{1,\ldots,r\\}$,
let $P^{K}$ be as before the set obtained from $P$ by removing the facets
$F_{i}$, $i\in K$. So $P^{\emptyset}=P$ and $P^{\\{1,\ldots,r\\}}={\rm
Int}(P)$. Let $\Sigma(1)=\\{\rho_{1},\ldots,\rho_{r}\\}$ be the rays of the
inner normal fan of $P$, and denote by $h=(h_{1},\ldots,h_{r})$ a vector of
real numbers indexed by the facets of $P$ (i.e., $h_{i}:=h_{\rho_{i}}$).
Consider the following operator in ${\mathbb{Q}}\\{\frac{\partial}{\partial
h_{i}}\mid i=1,\ldots,r\\}[y]$:

(232) $T_{y}^{K}(\frac{\partial}{\partial h}):=(1+y)^{n-r}\cdot\sum_{g\in
G_{\Sigma}}\prod_{i\in K}\frac{(1+y)\cdot
a_{\rho_{i}}(g)\cdot\frac{\partial}{\partial h_{i}}\cdot
e^{-\frac{\partial}{\partial h_{i}}}}{1-a_{\rho_{i}}(g)\cdot
e^{-\frac{\partial}{\partial h_{i}}}}\cdot\prod_{i\notin
K}\frac{\frac{\partial}{\partial h_{i}}(1+y\cdot
a_{\rho_{i}}(g)e^{-\frac{\partial}{\partial h_{i}}})}{1-a_{\rho_{i}}(g)\cdot
e^{-\frac{\partial}{\partial h_{i}}}},$

corresponding to formula (136) for the equivariant Todd class of
$mC_{y}^{\mathbb{T}}([U\hookrightarrow X])$, with $U=X\setminus D_{K}$ the
open complement of the divisor $D_{K}=\bigcup_{i\in K}D_{i}$ and
$D_{i}:=D_{\rho_{i}}$. By additivity,

${\rm td}_{*}^{\mathbb{T}}(mC_{y}^{\mathbb{T}}([U\hookrightarrow
X]))=T_{y*}([U\hookrightarrow X])=T_{y*}(X)-T_{y*}(D_{K})$

By Corollary 3.15, applied to $X$ and $D_{K}$, one gets a corresponding
equivariant Hirzebruch-Riemann-Roch formula:

(233) $\int_{X}{\rm td}_{*}^{\mathbb{T}}({\mathcal{O}}_{X}(D)\otimes
mC_{y}^{\mathbb{T}}([U\hookrightarrow X]))=\sum_{E\preceq
P^{K}}(1+y)^{\dim(E)}\cdot\sum_{m\in{\rm Relint}(E)\cap M}e^{s(m)}.$

With the above notations this then gives the following.

######  Theorem 6.4.

Let $P$ be a full-dimensional simple lattice polytope in $M_{\mathbb{R}}$, and
let $f$ be a polynomial function on $M_{\mathbb{R}}$. Then:

(234) $T_{y}^{K}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\ \sum_{E\preceq P^{K}}(1+y)^{\dim(E)}\cdot\sum_{m\in{\rm
Relint}(E)\cap M}f(m).$

By renormalization, $T_{y}^{K}(\frac{\partial}{\partial h})$ changes to

$\widehat{T}_{y}^{K}(\frac{\partial}{\partial h}):=\sum_{g\in
G_{\Sigma}}\prod_{i\in K}\frac{(1+y)\cdot
a_{\rho_{i}}(g)\cdot\frac{\partial}{\partial h_{i}}\cdot
e^{-(1+y)\frac{\partial}{\partial h_{i}}}}{1-a_{\rho_{i}}(g)\cdot
e^{-(1+y)\frac{\partial}{\partial h_{i}}}}\cdot\prod_{i\notin
K}\frac{\frac{\partial}{\partial h_{i}}(1+y\cdot
a_{\rho_{i}}(g)e^{-(1+y)\frac{\partial}{\partial
h_{i}}})}{1-a_{\rho_{i}}(g)\cdot e^{-(1+y)\frac{\partial}{\partial h_{i}}}}$

fitting with formula (137). We thus get by (224) the following

######  Corollary 6.5.

Let $P$ be a full-dimensional simple lattice polytope in $M_{\mathbb{R}}$, and
let $f$ be a polynomial function on $M_{\mathbb{R}}$. Then:

(235) $\widehat{T}_{y}^{K}(\frac{\partial}{\partial
h})\left(\int_{P_{y}(h)}f(m)\ dm\right)_{|_{h=0}}=\ \sum_{E\preceq
P^{K}}(1+y)^{\dim(E)}\cdot\sum_{m\in{\rm Relint}(E)\cap M}f((1+y)m).$

######  Remark 6.6.

In the corresponding local case of the tangent cone $Tan(P,v)$ of $P$ at a
vertex $v$, for $z\in N_{\mathbb{C}}$ small enough with $-z\in Int(\sigma)$,
we get by (155), (157) (172), together with (192), the following:

(236) $\begin{split}T^{K}_{y}(\frac{\partial}{\partial
h})\left(\int_{Tan(P,v)(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}=\sum_{E\preceq Tan(P,v)^{K}}(1+y)^{\dim(E)}\cdot
e^{\langle v,z\rangle}\cdot\sum_{m\in{\rm Relint}(E)\cap M}e^{\langle
m,z\rangle}\>.\end{split}$

We next discuss a weighted Euler-Maclaurin formula for a face of a simple
lattice polytope. Let $P$ be a full-dimensional simple lattice polytope in
$M_{\mathbb{R}}$, and fix a face $E$ of $P$. Let $\sigma:=\sigma_{E}$ be the
corresponding cone in the inner normal fan $\Sigma=\Sigma_{P}$ of $P$, with
$V_{\sigma}=V_{\sigma_{E}}=X_{E}$ the closure of the orbit of $\sigma$ in
$X=X_{P}$. Denote by $i_{E}=i_{\sigma}:V_{\sigma}\hookrightarrow X$ the closed
inclusion map. Then $V_{\sigma}$ is a simplicial toric variety whose fan is
$Star(\sigma)$, as defined in (28), which is built from cones $\tau\in\Sigma$
that have $\sigma$ as a face. Recall that ${\mathbb{T}}=T_{N}$ acts on
$V_{\sigma}$ via the morphism $T_{N}\to T_{N(\sigma)}$. We have by (225) the
following formula:

(237) $T_{y*}^{\mathbb{T}}([id_{V_{\sigma}}])=(1+y)^{n-r}\cdot
i_{\sigma}^{*}\left(\sum_{g\in G_{Star(\sigma)}}\prod_{\rho\in
Star(\sigma)(1)}\frac{F_{\rho}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}\right).$

The projection formula then yields the computation of the following
equivariant Hirzebruch class:

(238) $T_{y*}^{\mathbb{T}}([{V_{\sigma}}\hookrightarrow
X])=(1+y)^{n-r}\cdot\sum_{g\in G_{Star(\sigma)}}{\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}F_{\rho}\cdot\prod_{\rho\in
Star(\sigma)(1)}\frac{F_{\rho}\cdot\big{(}1+y\cdot a_{\rho}(g)\cdot
e^{-F_{\rho}}\big{)}}{1-a_{\rho}(g)\cdot e^{-F_{\rho}}}.$

Using Corollary 3.15, this yields the following result:

######  Theorem 6.7.

Let $P$ be a full-dimensional simple lattice polytope in $M_{\mathbb{R}}$, and
let $f$ be a polynomial function on $M_{\mathbb{R}}$. Then, for a fixed face
$E$ of $P$, with corresponding cone $\sigma$ in the inner normal fan $\Sigma$
of $P$, one has:

(239) $T_{y}^{E}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{E^{\prime}\preceq
E}(1+y)^{\dim(E^{\prime})}\cdot\sum_{m\in{\rm Relint}(E^{\prime})\cap M}f(m),$

where

(240) $T_{y}^{E}(\frac{\partial}{\partial h}):=(1+y)^{n-r}\cdot\sum_{g\in
G_{Star(\sigma)}}{\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}\frac{\partial}{\partial
h_{\rho}}\cdot\prod_{\rho\in Star(\sigma)(1)}\frac{\frac{\partial}{\partial
h_{\rho}}\left(1+y\cdot a_{\rho}(g)\cdot e^{-\frac{\partial}{\partial
h_{\rho}}}\right)}{1-a_{\rho}(g)\cdot e^{-\frac{\partial}{\partial
h_{\rho}}}}.$

For $y=0$, formula (239) specializes to (218). Taking the top degree in $y$ on
both sides of (239) gives back formula (214).

By renormalization, $T_{y}^{E}(\frac{\partial}{\partial h})$ changes to

$\widehat{T}_{y}^{E}(\frac{\partial}{\partial h}):=\sum_{g\in
G_{Star(\sigma)}}{\rm
mult}(\sigma)\cdot\prod_{\rho\in\sigma(1)}\frac{\partial}{\partial
h_{\rho}}\cdot\prod_{\rho\in Star(\sigma)(1)}\frac{\frac{\partial}{\partial
h_{\rho}}\left(1+y\cdot a_{\rho}(g)\cdot e^{-(1+y)\frac{\partial}{\partial
h_{\rho}}}\right)}{1-a_{\rho}(g)\cdot e^{-(1+y)\frac{\partial}{\partial
h_{\rho}}}}.$

We thus get by (224) the following

######  Corollary 6.8.

In the above notations, we get

(241) $\widehat{T}_{y}^{E}(\frac{\partial}{\partial
h})\left(\int_{P_{y}(h)}f(m)\ dm\right)_{|_{h=0}}=\sum_{E^{\prime}\preceq
E}(1+y)^{\dim(E^{\prime})}\cdot\sum_{m\in{\rm Relint}(E^{\prime})\cap
M}f((1+y)m).$

For $y=1$, one gets Euler-Maclaurin formulas corresponding to operators
related to suitable $L$-classes, i.e., $\widehat{T}_{1*}(X_{E})=L_{*}(X_{E})$
the Thom-Milnor $L$-class of $X_{E}$.

Another way to obtain examples of explicit weighted Euler-Maclaurin formulae
is by twisting the equivariant motivic Chern class $mC_{y}^{\mathbb{T}}$ by
sheaves of the form ${\mathcal{O}}_{X}(D^{\prime}-D)$, for $D=D_{P}$ the
original ample divisor associated to the full-dimensional lattice polytope,
and $D^{\prime}$ any ${\mathbb{T}}$-invariant Cartier divisor on $X$. In the
next example, we illustrate this principle in the case of
$mC_{y}^{\mathbb{T}}(X)$, corresponding to Theorem 6.2 and Corollary 6.3.

######  Example 6.9.

Let $D^{\prime}$ be a globally generated ${\mathbb{T}}$-invariant Cartier
divisor on $X$, with associated (not necessarily full-dimensional) lattice
polytope $P_{D^{\prime}}\subset M_{\mathbb{R}}$. Let
$D^{\prime}-D=\sum_{\rho\in\Sigma(1)}d_{\rho}D_{\rho}$ as a
${\mathbb{T}}$-invariant Cartier divisor. Let $X_{D^{\prime}}$ be the toric
variety of the lattice polytope $P_{D^{\prime}}$, defined via the
corresponding generalized fan as in [16][Prop.6.2.3]. By [16][Thm.6.2.8],
there is a proper toric morphism $f:X\to X_{D^{\prime}}$, induced by the
corresponding lattice projection $N\to N_{D^{\prime}}$ given by dividing out
by the minimal cone of the generalized fan of $P_{D^{\prime}}$. In particular,
$f:X\to X_{D^{\prime}}$ is a toric fibration. For $\sigma^{\prime}$ a cone in
the generalized fan of $P_{D^{\prime}}$, define as in (106)

$d_{\ell}(X/\sigma^{\prime}):=|\Sigma_{\ell}(X/\sigma^{\prime})|$

with

$\Sigma_{\ell}(X/\sigma^{\prime}):=\\{\sigma\in\Sigma\mid O_{\sigma}\subset
X,\ f(O_{\sigma})=O_{\sigma^{\prime}},\
\ell=\dim(O_{\sigma})-\dim(O_{\sigma^{\prime}})\\}.$

Consider the infinite order differential operators

(242) $T^{\prime}_{y}(\frac{\partial}{\partial
h}):=e^{\sum_{\rho\in\Sigma(1)}d_{\rho}\cdot\frac{\partial}{\partial
h_{\rho}}}\cdot T_{y}(\frac{\partial}{\partial h})$

and

(243) $\widehat{T}_{y}^{\prime}(\frac{\partial}{\partial
h}):=e^{(1+y)\cdot\sum_{\rho\in\Sigma(1)}d_{\rho}\cdot\frac{\partial}{\partial
h_{\rho}}}\cdot\widehat{T}_{y}(\frac{\partial}{\partial h}),$

with $T_{y}(\frac{\partial}{\partial
h}),\widehat{T}_{y}(\frac{\partial}{\partial h})$ as in Theorem 6.2 and
Corollary 6.3. For any quasi-polynomial $f$ on $M_{\mathbb{R}}$, one then has
by (122) the following new weighted Euler-Maclaurin formulae:

(244) $T^{\prime}_{y}(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\\\ =\sum_{E\preceq P_{D^{\prime}}}\left(\sum_{\ell\geq
0}(-1)^{\ell}\cdot
d_{\ell}(X/E)\cdot(1+y)^{\ell+\dim(E)}\right)\cdot\sum_{m\in{\rm
Relint}(E)\cap M}f(m).$ (245)
$\widehat{T}^{\prime}_{y}(\frac{\partial}{\partial
h})\left(\int_{P_{y}(h)}f(m)\ dm\right)_{|_{h=0}}=\\\ =\sum_{E\preceq
P_{D^{\prime}}}\left(\sum_{\ell\geq 0}(-1)^{\ell}\cdot
d_{\ell}(X/E)\cdot(1+y)^{\ell+\dim(E)}\right)\cdot\sum_{m\in{\rm
Relint}(E)\cap M}f((1+y)m).$

As noted in Example 5.28, $P_{D^{\prime}}$ is an ${\mathbb{N}}$-Minkowski
summand (in the sense of [16][Def.6.2.11]) of the original polytope $P$, see
[16][Cor.6.2.15]. ∎

##  7\. Euler-Maclaurin formulae via the Cappell-Shaneson algebra

###  7.1. Cappell-Shaneson algebra vs. completed equivariant cohomology ring

In this section we recall and provide generalizations of the Euler-Maclaurin
formula of Cappell-Shaneson [18, 43] and explain its connection to equivariant
toric geometry. The Euler-Maclaurin formula of Cappell-Shaneson [18, 43] does
not use a dilation $P(h)$ of the full-dimensional lattice polytope $P\subset
M_{{\mathbb{R}}}$. Instead, a summation of integrals over the faces $E$ of $P$
is used (compare also with [14][Sect.3.7]). More precisely, for any polynomial
function $f$ on $M_{\mathbb{R}}$ one has:

(246) $\sum_{m\in P\cap M}\;f(m)=\sum_{E\preceq
P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)dm\>,$

with
$p_{\sigma_{E}}(\partial_{i})\in\mbox{$\mathbb{Q}$}[[\partial_{1},\dots,\partial_{n}]]$
suitable infinite order differential operators with constant rational
coefficients in the partial derivatives with respect to the coordinates of the
vector space $M_{\mathbb{R}}$. Here the Lebesgue measure $dm$ on $E$ is
normalized so that the unit cube in the lattice $Span(E_{0})\cap M$ has volume
$1$, with $E_{0}:=E-m_{0}$ a translation of $E$ by a vertex $m_{0}\in E$.

The infinite order differential operators $p_{\sigma_{E}}(\partial_{i})$ are
defined through some relations in what is called in [34][Sect.6] the Cappell-
Shaneson algebra of $P$:

${\mathcal{A}}(P):=\mbox{$\mathbb{Q}$}[[\partial_{1},\dots,\partial_{n}]][U_{F}\mid\text{$F$
a facet of $P$}]/\sim$

with relations

(247) $U_{F_{1}}\cdots U_{F_{k}}=0\quad\text{for distinct $F_{i}$ with \
$F_{1}\cap\dots\cap F_{k}=\emptyset$},$

and

(248) $\frac{\partial}{\partial m}+\sum_{F}\;\langle m,n_{F}\rangle
U_{F}=0\quad\text{for all $m\in M$ (or a basis of $M$).}$

Here $\frac{\partial}{\partial m}$ is the differentiation in the direction of
$m$, with $n_{F}\in M$ the minimal lattice vector orthogonal to $F$ and
pointing into $P$. Let us rewrite this presentation of the algebra
${\mathcal{A}}(P)$ (given above in terms of the polytope $P$) into the
language of this paper using another description of the completed Stanley-
Reisner ring

${\mathbb{Q}}[[x_{\rho}\mid\rho\in\Sigma(1)]]/\sim_{SR}=:\widehat{SR_{\mathbb{Q}}(\Sigma)}$

of the associated toric variety $X_{P}$ and the inner normal fan
$\Sigma=\Sigma_{P}$. Note that the inner normal vector $n_{F}$ corresponds to
the generator $u_{\rho}$ of the ray $\rho\in\Sigma(1)$ corresponding to $F$.
For simplicity, we fix a basis $m_{1},\dots,m_{n}$ of
$M\simeq\mbox{$\mathbb{Z}$}^{n}$, with $t_{i}$ the corresponding coordinates
on $M_{\mathbb{K}}$ (for ${\mathbb{K}}={\mathbb{Q}},{\mathbb{R}}$) with
respect to this basis, so that $\frac{\partial}{\partial t_{i}}=\partial_{i}$
for $i=1,\dots,n$. Then

$\mbox{$\mathbb{Q}$}[t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]\simeq\mbox{$\mathbb{Q}$}[\partial_{1},\dots,\partial_{n},U_{F}\mid\text{$F$
a facet of $P$}]$

via $t_{i}\mapsto\partial_{i}$ and $x_{\rho}\mapsto U_{F}$, with $F$ the facet
of $P$ corresponding to $\rho\in\Sigma(1)$. The relation (247) corresponds to
the Stanly-Reisner relation $\sim_{SR}$, and the relation (248) translates
into

(249) $t_{i}+\sum_{\rho\in\Sigma(1)}\;\langle m_{i},n_{\rho}\rangle
x_{\rho}=0\quad\text{for $i=1,\dots,n$.}$

Denote by $\sim$ both relations together on
$\mbox{$\mathbb{Q}$}[t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]$.
These are also homogeneous for the usual grading doubled (i.e., with
$x_{\rho}$ and $t_{i}$ of degree two). Then one gets (see also [28][Lecture13,
Lem.2.2]):

######  Lemma 7.1.

The inclusion
$\mbox{$\mathbb{Q}$}[x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]\hookrightarrow\mbox{$\mathbb{Q}$}[t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]$
induces an isomorphism of graded rings

$SR_{\mathbb{Q}}(\Sigma)=\mbox{$\mathbb{Q}$}[x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]/\sim_{SR}\>\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mbox{$\mathbb{Q}$}[t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]/\sim$

and their formal power series completions

$\widehat{SR_{\mathbb{Q}}(\Sigma)}=\mbox{$\mathbb{Q}$}[[x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]]/\sim_{SR}\>\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mbox{$\mathbb{Q}$}[[t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]]/\sim\>,$

with the analytic Stanley-Reisner subring
$SR^{an}_{\mathbb{Q}}(\Sigma)=\mbox{$\mathbb{Q}$}\\{x_{\rho}\mid\text{$\rho\in\Sigma(1)$}\\}/\sim_{SR}$
identified with the image of the projection from
$\mbox{$\mathbb{Q}$}\\{t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}\\}$.

###### Proof.

Using the invertible variable transformation $x_{\rho}\mapsto
x_{\rho},t_{i}\mapsto t^{\prime}_{i}:=t_{i}+\sum_{\rho\in\Sigma(1)}\;\langle
m_{i},n_{\rho}\rangle x_{\rho}$ (for $i=1,\dots,n$ and $\rho\in\Sigma(1)$),
the claim is reduced to the corresponding inclusion

$\mbox{$\mathbb{Q}$}[x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]\hookrightarrow\mbox{$\mathbb{Q}$}[t^{\prime}_{1},\dots,t^{\prime}_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]\>,$

with $\sim$ given by $\sim_{SR}$ and $t^{\prime}_{i}=0$ for $i=1,\dots,n$. In
this case the results are obvious. ∎

######  Corollary 7.2.

Using the isomorphism

$s:Sym_{\mathbb{Q}}(M)\simeq(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}=H_{\mathbb{T}}^{*}(pt;\mbox{$\mathbb{Q}$});\
\ m\mapsto-c^{1}_{\mathbb{T}}({\mathbb{C}}_{\chi^{m}})$

together with ${\mathbb{Q}}[t_{1},\dots,t_{n}]=Sym_{\mathbb{Q}}(M)$, one gets
isomorphisms of $(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$-algebras

$SR_{\mathbb{Q}}(\Sigma)\simeq\mbox{$\mathbb{Q}$}[t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]/\sim\>\simeq(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}[x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]/\sim\>\simeq
H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$

and

$\widehat{SR_{\mathbb{Q}}(\Sigma)}\simeq\mbox{$\mathbb{Q}$}[[t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]]/\sim\>\simeq(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}[[x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]]/\sim\>\simeq\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\>,$

with $t_{i}\mapsto s(m_{i})=-c^{1}_{\mathbb{T}}({\mathbb{C}}_{\chi^{m_{i}}})$
for $i=i,\dots,n$, and $x_{\rho}\mapsto
F_{\rho}=[D_{\rho}]_{\mathbb{T}}=c^{1}_{\mathbb{T}}(\mathcal{O}(D_{\rho}))$
for $\rho\in\Sigma(1)$.

###### Proof.

It suffices to note that the relation (249) corresponds to (67). ∎

######  Remark 7.3.

Let $P\subset M_{{\mathbb{R}}}$ be a simple full-dimensional lattice polytope
with associated projective toric variety $X=X_{P}$ and the inner normal fan
$\Sigma=\Sigma_{P}$. Fix a basis $m_{1},\dots,m_{n}$ of
$M\simeq\mbox{$\mathbb{Z}$}^{n}$. There is an algebra homomorphism from the
Cappell-Shaneson algebra ${\mathcal{A}}(P)$ to the completed equivariant
cohomology ring

(250)
$\begin{split}{\mathcal{A}}(P)&=\mbox{$\mathbb{Q}$}[[\partial_{1},\dots,\partial_{n}]][U_{F}]/\sim\>\to\widehat{SR_{\mathbb{Q}}(\Sigma)}\simeq\
\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$});\>\\\ &U_{F}\mapsto
x_{\rho}\simeq
F_{\rho}=[D_{\rho}]_{\mathbb{T}}=c^{1}_{\mathbb{T}}(\mathcal{O}(D_{\rho}))\end{split}$

as a
$\mbox{$\mathbb{Q}$}[[\partial_{1},\dots,\partial_{n}]]\simeq\widehat{H}_{\mathbb{T}}^{*}(pt;\mbox{$\mathbb{Q}$})\simeq(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$-algebra
by $\partial_{i}\simeq t_{i}\simeq s(m_{i})=-c^{1}_{\mathbb{T}}(\chi^{m_{i}})$
for $i=1,\ldots,n$, and with the facet $F$ of $P$ corresponding to the ray
$\rho\in\Sigma(1)$. As we shall now explain, the algebra map (250) is in fact
an isomorphism. In abstract terms, the completions
$\widehat{SR_{\mathbb{Q}}(\Sigma)}\simeq\
\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})$ are completions of
connected integer graded commutative rings $R^{*}$ with respect to the maximal
ideal $R^{>0}$ given by positive degree elements, whereas ${\mathcal{A}}(P)$
uses the completion with respect to the maximal ideal $I_{\mathbb{T}}$ of
$\mbox{$\mathbb{Q}$}[\partial_{1},\dots,\partial_{n}]\simeq(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$.
As $(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}[U_{F}]$ is not a finitely
generated $(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$-module, the
completion functor with respect to $I_{\mathbb{T}}$ is only left exact. In
particular, there is an injective algebra homomorphism
${\mathcal{A}}(P)\hookrightarrow\left(\widehat{H^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})}\right)_{I_{\mathbb{T}}}$,
which factorizes (250). Using the relations defining ${\mathcal{A}}(P)$, one
gets as in [18][Proposition on pag. 888] or [43][6.1, pag. 621] that the image
of this monomorphism is the
$(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$-submodule generated
by the equivalence classes $[V_{E}]\in{\mathcal{A}}(P)$ associated to all
faces $E$ of $P$. This also shows that one gets a natural algebra homomorphism
${\mathbb{Q}}[[U_{F}]]\to{\mathcal{A}}(P)$ inducing algebra homomorphisms

$\widehat{H}^{\mathbb{T}}_{*}(X;{\mathbb{Q}})\to{\mathcal{A}}(P)\to(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}[[x_{\rho}]]/\sim.$

Since this composition is an isomorphism, the first arrow is injective and the
second is surjective. Using the fact that
$H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\simeq(\Lambda_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\otimes_{\mathbb{Q}}H^{*}(X;{\mathbb{Q}})$,
one also has that
$\left(\widehat{H^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})}\right)_{I_{\mathbb{T}}}\simeq(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}\otimes_{\mathbb{Q}}H^{*}(X;{\mathbb{Q}})$,
which is also generated as a
$(\widehat{\Lambda}_{\mathbb{T}})_{\mbox{$\mathbb{Q}$}}$-module by the
equivariant fundamental classes of the orbit closures
$[V_{\sigma_{E}}]_{\mathbb{T}}$ associated to all faces $E$ of $P$ (since
$H^{*}(X;{\mathbb{Q}})$ is generated as a ${\mathbb{Q}}$-vector space by the
fundamental classes $[V_{\sigma_{E}}]$). This shows that
${\mathcal{A}}(P)\simeq\left(\widehat{H^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})}\right)_{I_{\mathbb{T}}}$.
Finally, the natural algebra homomorphism
$\left(\widehat{H^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})}\right)_{I_{\mathbb{T}}}\to\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})$
is an isomorphism by [10][Prop.1.4]. Therefore, the homomorphism (250) is an
isomorphism.

###  7.2. Euler-Maclaurin formulae via the Cappell-Shaneson algebra

Let us now introduce the key functionals following [34][Section 6].

######  Definition 7.4.

Let

$p(t_{i},x_{\rho}):=\sum_{\alpha=(\alpha_{i})\in\mbox{$\mathbb{N}$}_{0}^{n}}\>p_{\alpha}(x_{\rho})\prod_{i=1}^{n}t_{i}^{\alpha_{i}}\in{\mathbb{Q}}\\{t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}\\},$

resp.,
${\mathbb{Q}}[[t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]]$ be a
convergent, resp., formal power series in the $t_{i},x_{\rho}$. Then

(251) $p(\partial_{i},\frac{\partial}{\partial h})\left(\int_{P(h)}e^{\langle
m,z\rangle}\
dm\right)_{|_{h=0}}:=\sum_{\alpha=(\alpha_{i})\in\mbox{$\mathbb{N}$}_{0}^{n}}\>p_{\alpha}(\frac{\partial}{\partial
h})\left(\int_{P(h)}\prod_{i=1}^{n}\partial_{i}^{\alpha_{i}}e^{\langle
m,z\rangle}\ dm\right)_{|_{h=0}}$

resp.,

(252) $p(\partial_{i},\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}:=\sum_{\alpha=(\alpha_{i})\in\mbox{$\mathbb{N}$}_{0}^{n}}\>p_{\alpha}(\frac{\partial}{\partial
h})\left(\int_{P(h)}\prod_{i=1}^{n}\partial_{i}^{\alpha_{i}}f(m)\
dm\right)_{|_{h=0}}$

for $f$ a polynomial function on $M_{\mathbb{R}}$. Note that in the second
case in the last term only finitely many summands are non-zero due to
Corollary 5.3.

Of course in (251) one has to explain in which sense the series is convergent,
as will be discussed in the following. As a consequence of Theorem 5.1 (with
$y=0$), Remark 5.2 and Corollary 5.11, we get by differentiation and
convergence the following.

######  Corollary 7.5.

Let $p(\partial_{i},\frac{\partial}{\partial
h})\in{\mathbb{Q}}\\{\partial_{1},\dots,\partial_{n},\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)\\}$ be an infinite order differential operator
with constant rational coefficients, i.e., obtained by substituting
$t_{i}\mapsto\partial_{i},x_{\rho}\mapsto\frac{\partial}{\partial h_{\rho}}$
into a convergent power series with rational coefficients
$p(t_{i},x_{\rho})\in{\mathbb{Q}}\\{t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)\\}$.
Then, in the above notations, we get for $z$ small enough and away from the
hyperplanes $\langle i_{\sigma}^{*}F_{\rho},z\rangle=0$ for each ray
$\rho\in\sigma(1)$ of $\sigma\in\Sigma(n)$ the following formula:

(253) $\begin{split}&p(\partial_{i},\frac{\partial}{\partial
h})\left(\int_{P(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}=\sum_{\alpha=(\alpha_{i})\in\mbox{$\mathbb{N}$}_{0}^{n}}\>p_{\alpha}(\frac{\partial}{\partial
h})\left(\int_{P(h)}\prod_{i=1}^{n}\partial_{i}^{\alpha_{i}}e^{\langle
m,z\rangle}\ dm\right)_{|_{h=0}}\\\
&=\sum_{\alpha=(\alpha_{i})\in\mbox{$\mathbb{N}$}_{0}^{n}}\>\prod_{i=1}^{n}\langle
m_{i},z\rangle^{\alpha_{i}}\left(\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\langle
p_{\alpha}(i_{\sigma}^{*}F_{\rho}),z\rangle\right)\\\
&=\langle\sum_{\sigma\in\Sigma(n)}\frac{i_{\sigma}^{*}\left(e^{(c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}\left(\sum_{\alpha=(\alpha_{i})\in\mbox{$\mathbb{N}$}_{0}^{n}}\>\prod_{i=1}^{n}s(m_{i})^{\alpha_{i}}p_{\alpha}(F_{\rho})\right)\right)}{Eu^{\mathbb{T}}_{X}(x_{\sigma})},z\rangle\\\
&=\langle\int_{X}e^{(c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}p(s(m_{i}),F_{\rho}),z\rangle\>.\end{split}$

If
$p_{n}\in{\mathbb{Q}}[\partial_{1},\dots,\partial_{n},\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)]$ is the corresponding truncation of $p$ up to
order $n$, then both sides of (253) applied to $p_{n}$ converge for
$n\to\infty$ locally uniformly in these $z$ to (253) applied to $p$.

###### Proof.

The second equality follows from
$\prod_{i=1}^{n}\partial_{i}^{\alpha_{i}}e^{\langle
m,z\rangle}=\prod_{i=1}^{n}\langle m_{i},z\rangle^{\alpha_{i}}e^{\langle
m,z\rangle}$ and Corollary 5.11. The third equality uses
$m_{i}=i_{\sigma}^{*}s(m_{i})\in M\simeq H^{2}_{\mathbb{T}}(pt;{\mathbb{Q}})$
for $s(m_{i})\in H^{2}_{\mathbb{T}}(X;{\mathbb{Q}})$. The last equality
follows from equation (177), where the element $p(s(m_{i}),F_{\rho})$
corresponds to the image of $p(t_{i},x_{\rho})$ under the evaluation
homomorphism

${\mathbb{Q}}\\{t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)\\}\mapsto\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}:\>t_{i}\mapsto
s(m_{i}),x_{\rho}\mapsto F_{\rho}\>.$

∎

######  Remark 7.6.

By (253), the operator $p(\partial_{i},\frac{\partial}{\partial
h})\left(\int_{P(h)}e^{\langle m,z\rangle}\ dm\right)_{|_{h=0}}$ depends only
on the equivalence class of $[p(t_{i},x_{\rho})]\in
SR^{an}_{\mathbb{Q}}(\Sigma)\simeq\mbox{$\mathbb{Q}$}\\{t_{1},\dots,t_{n},x_{\rho}\mid\text{$\rho\in\Sigma(1)$}]\\}/\sim$
and not on the chosen convergent power series representative.

######  Remark 7.7.

Assume in addition in Corollary 7.5, that
$\langle\int_{X}e^{(c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}p(s(m_{i}),F_{\rho}),z\rangle$
is a convergent power series in $z$ near zero. Then one gets as an application
of Cauchy’s integral formula (see also [34][p.27]), that both sides of (253)
applied to $p_{n}$ converge for $n\to\infty$ and $z$ small locally uniformly
to (253) applied to $p$. In particular, this limit commutes with finite order
differentiations with respect to $z$ (and $z$ small enough).

######  Corollary 7.8.

Let $p(\partial_{i},\frac{\partial}{\partial
h})\in{\mathbb{Q}}[[\partial_{1},\dots,\partial_{n},\frac{\partial}{\partial
h_{\rho}}\mid\rho\in\Sigma(1)]]$ be an infinite order differential operator
with constant rational coefficients, i.e., obtained by substituting
$t_{i}\mapsto\partial_{i},x_{\rho}\mapsto\frac{\partial}{\partial h_{\rho}}$
into a formal power series with rational coefficients
$p(t_{i},x_{\rho})\in{\mathbb{Q}}[[t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)]]$.
Then for a polynomial function $f$ on $M_{\mathbb{R}}$, we have the following
formula:

(254) $p(\partial_{i},\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=f(\frac{\partial}{\partial
z})\left(\langle\int_{X}e^{c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}p(s(m_{i}),F_{\rho}),z\rangle\right)_{|_{z=0}}\>,$

where on the right hand side the operator $\left(f(\frac{\partial}{\partial
z})\right)_{|_{z=0}}$ acts on a formal power series in $z$.

###### Proof.

The proof is similar to that of Corollary 5.14. ∎

######  Remark 7.9.

By (254), the operator $p(\partial_{i},\frac{\partial}{\partial
h})\left(\int_{P(h)}f(z)\ dm\right)_{|_{h=0}}$ depends only on the equivalence
class of $[p(t_{i},x_{\rho})]\in\widehat{SR_{\mathbb{Q}}(\Sigma)}$ and not on
the chosen formal power series representative.

Altogether, we get the following second abstract Euler-Maclaurin formula
coming from the equivariant Hirzebruch-Riemann-Roch theorem.

######  Theorem 7.10.

Let $X=X_{P}$ be the projective simplicial toric variety associated to a full-
dimensional simple lattice polytope $P\subset M_{\mathbb{R}}$. Let
$\Sigma:=\Sigma_{P}$ be the inner normal fan of $P$, and $D:=D_{P}$ the ample
Cartier divisor associated to $P$. Let $[{\mathcal{F}}]\in
K^{\mathbb{T}}_{0}(X)$ be fixed, and choose a convergent power series
$p(t_{i},x_{\rho})\in{\mathbb{Q}}\\{t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)\\}$
so that $p(s(m_{i}),F_{\rho})={\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}])\in\left(H^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\right)^{an}$.
Then

(255) $\begin{split}p(\partial_{i},\frac{\partial}{\partial
h})\left(\int_{P(h)}e^{\langle m,z\rangle}\
dm\right)_{|_{h=0}}&=\langle\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}),z\rangle\\\
&=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
e^{\langle m,z\rangle}\>,\end{split}$

as analytic functions in $z$ with $z$ small enough, and with
$\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})\in(\Lambda^{an}_{\mathbb{T}})_{\mathbb{Q}}$
the cohomological equivariant Euler characteristic of
${\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}$.

###### Proof.

Equation (253) can now be calculated as

$\displaystyle p(\partial_{i},\frac{\partial}{\partial
h})\left(\int_{P(h)}e^{\langle m,z\rangle}\ dm\right)_{|_{h=0}}$
$\displaystyle=$
$\displaystyle\langle\int_{X}e^{c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D))}{\rm
td}^{\mathbb{T}}_{*}([{\mathcal{F}}]),z\rangle$ $\displaystyle=$
$\displaystyle\langle\chi^{\mathbb{T}}(X,{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}}),z\rangle,$

where the last equality follows from the equivariant Hirzebruch-Riemann-Roch
formula (83) as in the proof of Theorem 5.18. Finally, in the proof we first
need to assume that $z$ is small enough and away from the hyperplanes $\langle
i_{\sigma}^{*}F_{\rho},z\rangle=0$ for each ray $\rho\in\sigma(1)$ of
$\sigma\in\Sigma(n)$, since the localization formula is used; however, formula
(255) then holds for all $z$ small enough, by Remark 7.7. ∎

Note that Theorem 7.10 reduces to Theorem 5.18 in the case when

$p(t_{i},x_{\rho})\in{\mathbb{Q}}\\{x_{\rho}\mid\rho\in\Sigma(1)\\}\subset{\mathbb{Q}}\\{t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)\\}$

does not depend on the variables $t_{i}$. Similarly, the next Corollary
reduces in this case to Corollary 5.21.

######  Corollary 7.11.

Let $X=X_{P}$ be the projective simplicial toric variety associated to a full-
dimensional simple lattice polytope $P\subset M_{\mathbb{R}}$. Let
$\Sigma:=\Sigma_{P}$ be the inner normal fan of $P$, and $D:=D_{P}$ the ample
Cartier divisor associated to $P$. Let $[{\mathcal{F}}]\in
K^{\mathbb{T}}_{0}(X)$ be fixed, and choose a formal power series
$p(t_{i},x_{\rho})\in{\mathbb{Q}}[[t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)]]$
so that
$p(s(m_{i}),F_{\rho})=td^{\mathbb{T}}_{*}([{\mathcal{F}}])\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$.
Then for a polynomial function $f$ on $M_{\mathbb{R}}$, we have:

(256) $p(\partial_{i},\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
f(m)\>.$

###### Proof.

This follows from Corollary 7.8 by applying the operator
$\left(f(\frac{\partial}{\partial z})\right)_{|_{z=0}}$ to the last term of
formula (255), viewed as a formal power series in $z$. ∎

For the use of integration over the faces $E$ of $P$ instead of dilatation of
the facets of $P$ by a parameter $h_{\rho}$ ($\rho\in\Sigma(1)$), we next
prove an analogue of Theorem 5.1 for faces of a polytope.

Assume $P$ is a full-dimensional simple lattice polytope in $M_{\mathbb{R}}$
with associated toric variety $X=X_{P}$ and inner normal fan
$\Sigma=\Sigma_{P}$ . Let $P(h)$ be the dilatation of $P$ with respect to the
vector $h=(h_{\rho})_{\rho\in\Sigma(1)}$ with real entries indexed by the rays
of $\Sigma$. So, if $P$ is defined by inequalities of the form

$\langle m,u_{\rho}\rangle+c_{\rho}\geq 0,$

with $u_{\rho}$ the ray generators and $c_{\rho}\in{\mathbb{Z}}$, for each
$\rho\in\Sigma(1)$, then $P(h)$ is defined by inequalities

$\langle m,u_{\rho}\rangle+c_{\rho}+h_{\rho}\geq 0,$

for each $\rho\in\Sigma(1)$. Similarly, we consider the dilatation $E(h)$ of a
fixed face $E$ of $P$, and let $\sigma_{E}\in\Sigma$ be the cone corresponding
to $E$.

######  Theorem 7.12.

In the above notations, we have

(257) $\begin{split}\int_{E(h)}e^{\langle m,z\rangle}\ dm&={\rm
mult}(\sigma_{E})\cdot\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P(h)})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\prod_{\rho\in\sigma_{E}(1)}\langle
i_{\sigma}^{*}F_{\rho},z\rangle\\\ &={\rm
mult}(\sigma_{E})\cdot\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle(i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot e^{\sum_{\rho}h_{\rho}\langle
i_{\sigma}^{*}F_{\rho},z\rangle}\cdot\prod_{\rho\in\sigma_{E}(1)}\langle
i_{\sigma}^{*}F_{\rho},z\rangle\>.\end{split}$

###### Proof.

The proof is similar to that of Theorem 5.1, with the following modifications.
Instead of ${\rm td}_{*}^{\mathbb{T}}(X)$, we use ${\rm
td}_{*}^{\mathbb{T}}([(i_{E})_{*}{\mathcal{O}}_{V_{\sigma_{E}}}])$, which is
given by formula (217), together with the corresponding equivariant Riemann-
Roch formula of (121) which is used for the calculation of Riemann sums
approximating the integral. The Lebesgue measure $dm$ on $E(h)$ is now
normalized so that the unit cube in the lattice $Span(E_{0})\cap M$ has volume
$1$, with $E_{0}:=E-m_{0}$ a translation of $E$ by a vertex $m_{0}\in E$. As
before, $\langle Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle$ is a homogeneous
polynomial of degree $n$ in $z$, but the additional factor
$\prod_{\rho\in\sigma_{E}(1)}\langle i_{\sigma}^{*}F_{\rho},z\rangle$ is
homogeneous of degree $\hbox{\rm codim}\,(E)$, fitting with the use of the
multiplication factor $\frac{1}{k^{\dim(E)}}$ needed for the Riemann sums
considered here. Finally, we have

(258) $\lim_{k\to\infty}\left(k^{\hbox{\rm codim}\,(E)}\cdot\langle
i^{*}_{\sigma}({\rm
td}_{*}^{\mathbb{T}}([(i_{E})_{*}{\mathcal{O}}_{V_{\sigma_{E}}}])),\frac{1}{k}\cdot
z\rangle\right)={\rm mult}(\sigma_{E})\cdot\prod_{\rho\in\sigma_{E}(1)}\langle
i_{\sigma}^{*}F_{\rho},z\rangle.$

This follows from formula (217) by the same calculation as in the proof of
Theorem 5.1. ∎

By combining the results of Theorem 5.1 and Theorem 7.12, we obtain as an
application a formula which relates integration over faces with a
differentiation with respect to the corresponding $h$’s.

######  Corollary 7.13.

Let $f$ be a polynomial on $M_{\mathbb{R}}$. Then

(259) $\int_{E(h)}f(m)e^{\langle m,z\rangle}\ dm={\rm
mult}(\sigma_{E})\cdot\prod_{\rho\in\sigma_{E}(1)}\frac{\partial}{\partial
h_{\rho}}\int_{P(h)}f(m)e^{\langle m,z\rangle}\ dm\>,$

as analytic functions in $z$ and $h$ near zero. In particular,

  * (a)

(260) $p(\frac{\partial}{\partial h})\left(\int_{E(h)}f(m)e^{\langle
m,z\rangle}\ dm\right)_{|_{h=0}}={\rm
mult}(\sigma_{E})\cdot\prod_{\rho\in\sigma_{E}(1)}\frac{\partial}{\partial
h_{\rho}}\cdot p(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)e^{\langle
m,z\rangle}\ dm\right)_{|_{h=0}}\>,$

for $p(x_{\rho})$ a polynomial in the variables $x_{\rho}$, $\rho\in\Sigma(1)$
(or a convergent power series as in the context of Remark 5.13).

  * (b)

(261) $p(\frac{\partial}{\partial h})\left(\int_{E(h)}f(m)\
dm\right)_{|_{h=0}}={\rm
mult}(\sigma_{E})\cdot\prod_{\rho\in\sigma_{E}(1)}\frac{\partial}{\partial
h_{\rho}}\cdot p(\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}\>,$

for a formal power series $p(x_{\rho})$ in the variables $x_{\rho}$,
$\rho\in\Sigma(1)$.

###### Proof.

For $f=1$, formula (259) follows by comparing the results of Theorem 5.1 and
Theorem 7.12. In the general case, we apply the operator
$f(\frac{\partial}{\partial z})$ to both sides of the formula obtained for
$f=1$. In (b) we finally evaluate the result at $z=0$. By Corollary 5.3, a
formal power series (as opposed to a convergent one) can be used in (261). ∎

In particular,

(262) $\frac{\partial}{\partial h_{\rho_{1}}}\cdots\frac{\partial}{\partial
h_{\rho_{k}}}\int_{P(h)}f(m)e^{\langle m,z\rangle}\ dm\>=0$

if the corresponding facets for different $\rho_{i}$ ($i=1,\dots,k$) do not
intersect.

As another application of Corollary 7.13, we can mention the following
reformulation of the classical Euler-Maclaurin formula (204) of Brion-Vergne
(using (35)), for $f$ a polynomial function on $M_{\mathbb{R}}$ (see also
[34][page 22]):

(263) $\begin{split}&\sum_{m\in P\cap M}f(m)=\sum_{E\preceq P}\sum_{g\in
G_{\sigma_{E}}^{\circ}}\prod_{\rho\in\Sigma(1)}\frac{\frac{\partial}{\partial
h_{\rho}}}{1-a_{\rho}(g)e^{-\frac{\partial}{\partial
h_{\rho}}}}\left(\int_{P(h)}f(m)\ dm\right)_{|_{h=0}}\\\ &=\sum_{E\preceq
P}\frac{1}{{\rm mult}(\sigma_{E})}\sum_{g\in
G_{\sigma_{E}}^{\circ}}\prod_{\rho\not\in\sigma_{E}(1)}\frac{\frac{\partial}{\partial
h_{\rho}}}{1-e^{-\frac{\partial}{\partial
h_{\rho}}}}\prod_{\rho\in\sigma_{E}(1)}\frac{1}{1-a_{\rho}(g)\cdot
e^{-\frac{\partial}{\partial h_{\rho}}}}\left(\int_{E(h)}f(m)\
dm\right)_{|_{h=0}}\>.\end{split}$

The Euler-Maclaurin formula using the last operator is due to Guillemin
[31][Thm.1.3 and Eqn.3.28]

As a further application of Theorem 7.12, we give an algebro-geometric proof
of the Stokes’ formula for polytopes (see also [34][Prop.6.1]).

######  Theorem 7.14.

Let $P\subset M_{\mathbb{R}}$ be a full-dimensional simple lattice polytope,
with corresponding inner fan $\Sigma$. Let $E_{\rho}$ denote the facet of $P$
corresponding to the ray $\rho\in\Sigma(1)$. For a fixed $m_{0}\in M$, let
$\frac{\partial}{\partial m_{0}}$ be the differentiation in direction of
$m_{0}$. Then,

(264) $\int_{P(h)}\frac{\partial}{\partial m_{0}}e^{\langle m,z\rangle}\
dm=-\sum_{\rho\in\Sigma(1)}\langle
m_{0},u_{\rho}\rangle\cdot\int_{E_{\rho}(h)}e^{\langle m,z\rangle}\ dm.$

###### Proof.

Recall that

$s(m_{0})=-c^{1}_{\mathbb{T}}({\mathbb{C}}_{\chi^{m_{0}}})=-\sum_{\rho\in\Sigma(1)}\langle
m_{0},u_{\rho}\rangle\cdot F_{\rho}\in H^{*}_{\mathbb{T}}(X;{\mathbb{Q}}).$

Then

$\begin{split}\int_{P(h)}\frac{\partial}{\partial m_{0}}e^{\langle
m_{,}z\rangle}\ dm&=\int_{P(h)}\langle m_{0},z\rangle\cdot e^{\langle
m,z\rangle}\ dm\\\
&\overset{\eqref{f113}}{=}\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P(h)})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\langle m_{0},z\rangle\\\
&=\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P(h)})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\langle
i_{\sigma}^{*}s(m_{0}),z\rangle\\\ &=-\sum_{\rho\in\Sigma(1)}\langle
m_{0},u_{\rho}\rangle\cdot\sum_{\sigma\in\Sigma(n)}\frac{e^{\langle
i_{\sigma}^{*}c^{\mathbb{T}}_{1}({\mathcal{O}}_{X}(D_{P(h)})),z\rangle}}{\langle
Eu^{\mathbb{T}}_{X}(x_{\sigma}),z\rangle}\cdot\langle
i_{\sigma}^{*}F_{\rho},z\rangle\\\
&\overset{\eqref{f113b}}{=}-\sum_{\rho\in\Sigma(1)}\langle
m_{0},u_{\rho}\rangle\cdot\int_{E_{\rho}(h)}e^{\langle m,z\rangle}\
dm,\end{split}$

as desired. ∎

Applying the operator $f(\frac{\partial}{\partial z})$ to formula (264) for
$f$ a polynomial function on $M_{\mathbb{R}}$, we get the following:

(265) $\int_{P(h)}\frac{\partial}{\partial m_{0}}f(m)e^{\langle m,z\rangle}\
dm=-\sum_{\rho\in\Sigma(1)}\langle
m_{0},u_{\rho}\rangle\cdot\int_{E_{\rho}(h)}f(m)e^{\langle m,z\rangle}\ dm$

as analytic functions in $h$ and $z$ near zero. Further evaluation at $z=0$
yields the identity

(266) $\int_{P(h)}\frac{\partial}{\partial m_{0}}f(m)\
dm=-\sum_{\rho\in\Sigma(1)}\langle
m_{0},u_{\rho}\rangle\cdot\int_{E_{\rho}(h)}f(m)\ dm,$

as polynomial functions in $h$ near zero.

######  Remark 7.15.

By Remark 7.3, there is a surjection
${\mathcal{A}}(P)\to\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$, as
$(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}$-modules, with
${\mathcal{A}}(P)$ generated by the equivalence classes of
$[V_{E}]\in{\mathcal{A}}(P)$, which are mapped to the equivariant fundamental
classes $[V_{\sigma_{E}}]_{\mathbb{T}}$ of orbit closures corresponding to all
faces $E$ of $P$. Hence, $\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$ is
generated as a $(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}$-module by the
equivariant fundamental classes $[V_{\sigma_{E}}]_{\mathbb{T}}$.

Let $[{\mathcal{F}}]\in K^{\mathbb{T}}_{0}(X)$ be fixed, and choose elements
$p_{\sigma_{E}}(t_{i})\in\widehat{H}^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})=(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}\simeq{\mathbb{Q}}[[t_{1},\dots,t_{n}]]$
with

(267) ${\rm td}_{*}^{\mathbb{T}}([{\mathcal{F}}])=\sum_{E\preceq
P}\>p_{\sigma_{E}}(t_{i})[V_{\sigma_{E}}]_{\mathbb{T}}\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\>.$

Then ${\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])=p(s(m_{i}),F_{\rho})\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$
for

(268) $p(t_{i},x_{\rho}):=\sum_{E\preceq P}\>\left({\rm
mult}(\sigma_{E})\cdot\prod_{\rho\in\sigma_{E}(1)}x_{\rho}\right)\cdot
p_{\sigma_{E}}(t_{i})\in{\mathbb{Q}}[[t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)]]\>.$

Altogether, with these notations, we get our third and final abstract Euler-
Maclaurin formula coming from the equivariant Hirzebruch-Riemann-Roch theorem,
which provides a generalization of the Cappell-Shaneson Euler-Maclaurin
formula (see Remark 7.17 below).

######  Theorem 7.16.

Let $X=X_{P}$ be the projective simplicial toric variety associated to a full-
dimensional simple lattice polytope $P\subset M_{\mathbb{R}}$. Let
$\Sigma:=\Sigma_{P}$ be the inner normal fan of $P$, and $D:=D_{P}$ the ample
Cartier divisor associated to $P$. Let $[{\mathcal{F}}]\in
K^{\mathbb{T}}_{0}(X)$ be fixed, and choose the formal power series
$p(t_{i},x_{\rho})\in{\mathbb{Q}}[[t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)]]$
as in (268). Then for a polynomial function $f$ on $M_{\mathbb{R}}$, we have:

(269) $\sum_{E\preceq P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)\
dm=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}(D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
f(m)\>.$

###### Proof.

The assertion follows from Corollary 7.13 by the following calculation:

(270) $\begin{split}&p(\partial_{i},\frac{\partial}{\partial
h})\left(\int_{P(h)}f(m)\ dm\right)_{|_{h=0}}\\\ &=\sum_{E\preceq
P}\>\left({\rm
mult}(\sigma_{E})\cdot\prod_{\rho\in\sigma_{E}(1)}\frac{\partial}{\partial
h_{\rho}}\right)\left(\int_{P(h)}p_{\sigma_{E}}(\partial_{i})f(m)\
dm\right)_{|_{h=0}}\\\ &=\sum_{E\preceq
P}\>\left(\int_{E(h)}p_{\sigma_{E}}(\partial_{i})f(m)\ dm\right)_{|_{h=0}}\\\
&=\sum_{E\preceq P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)\
dm\>.\end{split}$

Here the first equality follows from the definition (252) of
$p(\partial_{i},\frac{\partial}{\partial h})\left(\int_{P(h)}f(m)\
dm\right)_{|_{h=0}}$ and the choice (268) of $p(t_{i},x_{\rho})$. Then the
second equality follows from (261), and the final equality is just continuity
in $h$. ∎

######  Remark 7.17.

In the classical case ${\mathcal{F}}:={\mathcal{O}}_{X}$, this is just
Cappell-Shaneson’s recipe for the definition of the differential operators
$p_{\sigma_{E}}(\partial_{i})$, described here geometrically in terms of the
equivariant Todd class ${\rm td}_{*}^{\mathbb{T}}(X):={\rm
td}_{*}^{\mathbb{T}}([{\mathcal{O}}_{X}])\in\widehat{H}^{*}_{\mathbb{T}}(X;\mbox{$\mathbb{Q}$})$
(see [18][Theorem 2]). More precisely, the differential operator
$\mathcal{E}=\sum_{E\preceq P}P_{E}\cdot
U_{E}\in{\mathbb{Q}}[[\partial_{1},\dots,\partial_{n}]][U_{F}]$ used in
[43][6.2, page 622] and [18][page 888] maps under the homomorphism (250) to a
presentation of the equivariant Todd class ${\rm
td}_{*}^{\mathbb{T}}(X)=\sum_{E\preceq
P}\>p_{\sigma_{E}}(t_{i})[V_{\sigma_{E}}]_{\mathbb{T}}\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$.
Indeed, the classes $[V_{\sigma_{E}}]_{\mathbb{T}}$ are the images of the
elements $U_{E}\in{\mathbb{Q}}[[\partial_{1},\dots,\partial_{n}]][U_{F}]$ used
in loc.cit., and the operators
$P_{E}\in{\mathbb{Q}}[[\partial_{1},\dots,\partial_{n}]]$ are deduced in [18,
43] from a representative
$T(E)\in{\mathbb{Q}}[[\partial_{1},\dots,\partial_{n}]][U_{F}]$ mapping to the
equivariant Todd class ${\rm td}_{*}^{\mathbb{T}}(X)$ calculated via the
equivariant version of Theorem 2.6. So we can choose $p_{\sigma_{E}}$ to be
the images of these $P_{E}$ in $(\widehat{\Lambda}_{\mathbb{T}})_{\mathbb{Q}}$
under (250). Hence, the presentation

$p(t_{i},x_{\rho}):=\sum_{E\preceq P}\>\left({\rm
mult}(\sigma_{E})\cdot\prod_{\rho\in\sigma_{E}(1)}x_{\rho}\right)\cdot
p_{\sigma_{E}}(t_{i})\in{\mathbb{Q}}[[t_{1},\dots,t_{n},x_{\rho}\mid\rho\in\Sigma(1)]]$

chosen here is exactly the image of $\mathcal{E}$ in
${\mathbb{Q}}[[t_{1},\dots,t_{n}]][[x_{\rho}]]$.

In this case, when ${\mathcal{F}}:={\mathcal{O}}_{X}$, the right hand side of
(269) becomes $\sum_{m\in P\cap M}f(m)$ by Example 5.23. Specializing further
to $f=1$ yields then the classical formula (see, e.g., [27][page 112]):

(271) $|P\cap M|=\sum_{E\preceq P}r_{E}\cdot vol(E)\>,$

with $r_{E}=p_{\sigma_{E}}(0)$ the constant coefficient of $p_{\sigma_{E}}$
and

(272) ${\rm td}_{*}(X)=\sum_{E\preceq P}\>r_{E}[V_{\sigma}]\in
H^{*}(X;{\mathbb{Q}})$

the non-equivariant version of

(273) ${\rm td}_{*}^{\mathbb{T}}([{\mathcal{O}}_{X}])=\sum_{E\preceq
P}\>p_{\sigma_{E}}(t_{i})[V_{\sigma_{E}}]_{\mathbb{T}}\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\>.$

When ${\mathcal{F}}:=\omega_{X}$, the right hand side of (269) becomes
$\sum_{m\in{\rm Int}(P)\cap M}f(m)$ by Example 5.24. Moreover, here we can
choose

(274) ${\rm td}_{*}^{\mathbb{T}}([\omega_{X}])=\sum_{E\preceq
P}\>p_{\sigma_{E}}(-t_{i})\cdot(-1)^{\hbox{\rm
codim}\,(E)}[V_{\sigma_{E}}]_{\mathbb{T}}\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})\>,$

by Remark 3.26, with $p_{\sigma_{E}}(t_{i})$ as in (273). Specializing further
to $f=1$ yields the formula

(275) $|{\rm Int}(P)\cap M|=\sum_{E\preceq P}(-1)^{\hbox{\rm codim}\,(E)}\cdot
r_{E}\cdot vol(E)\>,$

with $r_{E}$ as in (272).

######  Example 7.18.

Here we list further specializations of formula (269) for appropriate choices
of $[{\mathcal{F}}]\in K^{\mathbb{T}}_{0}(X)$, resp., $[{\mathcal{F}}]\in
K^{\mathbb{T}}_{0}(X)[y]$, with ${\rm
td}_{*}^{\mathbb{T}}([{\mathcal{F}}])=\sum_{E\preceq
P}\>p_{\sigma_{E}}(t_{i})[V_{\sigma_{E}}]_{\mathbb{T}}\in\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})$,
resp., $\widehat{H}^{*}_{\mathbb{T}}(X;{\mathbb{Q}})[y]$ for some elements
$p_{\sigma_{E}}(t_{i})\in\widehat{H}^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})$,
resp., $\widehat{H}^{*}_{\mathbb{T}}(pt;{\mathbb{Q}})[y]$.

  1. (1)

For $[{\mathcal{F}}]=mC_{y}^{\mathbb{T}}(X)\in K^{\mathbb{T}}_{0}(X)[y]$,
(269) becomes by equation (230):

$\sum_{E\preceq P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)\
dm=\sum_{E^{\prime\prime}\preceq
P}(1+y)^{\dim(E^{\prime\prime})}\sum_{m\in{\rm Relint}(E^{\prime\prime})\cap
M}f(m)\>.$

  2. (2)

For $[{\mathcal{F}}]=mC_{0}^{\mathbb{T}}([U\hookrightarrow X])\in
K^{\mathbb{T}}_{0}(X)$, with $U=X\setminus D_{K}$ the open complement of the
divisor $D_{K}=\bigcup_{i\in K}D_{i}$ and $D_{i}:=D_{\rho_{i}}$, (269) becomes
by equation (210):

$\sum_{E\preceq P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)\ dm=\sum_{m\in
P^{K}\cap M}f(m)\>.$

Here $P^{K}$ is the set obtained from $P$ by removing the facets $F_{i}$ for
$i\in K$. More generally, for
$[{\mathcal{F}}]=mC_{y}^{\mathbb{T}}([U\hookrightarrow X])\in
K^{\mathbb{T}}_{0}(X)[y]$, (269) becomes by equation (234):

$\sum_{E\preceq P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)\
dm=\sum_{E^{\prime\prime}\preceq
P^{K}}(1+y)^{\dim(E^{\prime\prime})}\cdot\sum_{m\in{\rm
Relint}(E^{\prime\prime})\cap M}f(m)\>.$

  3. (3)

For $[{\mathcal{F}}]=[(i_{\sigma})_{*}\omega_{V_{\sigma}}]\in
K^{\mathbb{T}}_{0}(X)$, with $V_{\sigma}=V_{\sigma_{E^{\prime}}}$ the closure
of the orbit of $\sigma$ in $X=X_{P}$ corresponding to the face $E^{\prime}$
of $P$, (269) becomes by equation (214):

$\sum_{E\preceq P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)\
dm=\sum_{m\in{\rm Relint}(E^{\prime})\cap M}f(m)\>.$

  4. (4)

For $[{\mathcal{F}}]=[(i_{\sigma})_{*}{\mathcal{O}}_{V_{\sigma}}]\in
K^{\mathbb{T}}_{0}(X)$, with $V_{\sigma}=V_{\sigma_{E^{\prime}}}$ the closure
of the orbit of $\sigma$ in $X=X_{P}$ corresponding to the face $E^{\prime}$
of $P$, (269) becomes by equation (218):

$\sum_{E\preceq P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)\ dm=\sum_{m\in
E^{\prime}\cap M}f(m)\>.$

More generally, for
$[{\mathcal{F}}]=(i_{\sigma})_{*}mC_{y}^{\mathbb{T}}(V_{\sigma})\in
K^{\mathbb{T}}_{0}(X)[y]$, (269) becomes by equation (239):

$\sum_{E\preceq P}\>\int_{E}p_{\sigma_{E}}(\partial_{i})f(m)\
dm=\sum_{E^{\prime\prime}\preceq
E^{\prime}}(1+y)^{\dim(E^{\prime\prime})}\cdot\sum_{m\in{\rm
Relint}(E^{\prime\prime})\cap M}f(m)\>.$

∎

###  7.3. Generalized Reciprocity for Dedekind Sums via Euler-Maclaurin
formulae

We conclude this paper with the following application of formula (269), with
$D$ replaced by $(1+y)D$.

######  Corollary 7.19.

In the context of Theorem 7.16, we have an equality of polynomials in $1+y$,

(276) $\sum_{E_{y}\preceq
P_{y}}\>\int_{E_{y}}p_{\sigma_{E_{y}}}(\partial_{i})f(m)\ dm=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{O}}_{X}((1+y)D)\otimes{\mathcal{F}})_{\chi^{-m}}\right)\cdot
f(m)\>.$

When evaluating these polynomials at zero (i.e., for $y=-1$), one gets the
following identity:

(277) $\sum_{v\in P}\left(p_{\sigma_{v}}(\partial_{i})f\right)(0)=\sum_{m\in
M}\left(\sum_{i=0}^{n}(-1)^{i}\cdot\dim_{\mathbb{C}}H^{i}(X;{\mathcal{F}})_{\chi^{-m}}\right)\cdot
f(m).$

where the left hand sum is over the vertices of $P$.

###### Proof.

The right hand sides of the above expressions are calculated in Corollary 5.4.
For the left hand side of (276), note that for a given face $E$ of $P$, if $f$
is a homogeneous polynomial of degree $d_{f}$, and $p(\partial_{i})$ is a
homogeneous differential operator of degree $d\leq d_{f}$, then

(278) $\int_{E_{y}}p(\partial_{i})f(m)\
dm=(1+y)^{\dim(E)+d_{f}-d}\int_{E}p(\partial_{i})f(m)\ dm.$

This explains the polynomial behavior of the left hand side of (276), and it
yields (277) upon evaluation at zero (since by letting $y=-1$, all terms on
the right hand side of (278) vanish except for $\dim(E)=0$ and $d=d_{f}$). ∎

######  Example 7.20.

If ${\mathcal{F}}={\mathcal{O}}_{X}$ in (277), one gets by Example 3.18 the
following identity:

(279) $\sum_{v\in P}\left(p_{\sigma_{v}}(\partial_{i})f\right)(0)=f(0).$

For instance, in the case of lattice polygones, this formula yields
generalizations of reciprocity laws for classical Dedekind sums (using, e.g.,
the explicit description of the operators $p_{\sigma_{v}}(\partial_{i})$ from
[18][page 889]). Details will be explained in forthcoming work. ∎

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