# The Nielsen realization problem for K3 surfaces

Benson Farb and Eduard Looijenga The first author was supported in part by
National Science Foundation Grant No. DMS-181772 and the Eckhardt Faculty
Fund. The second author is supported by the Chinese National Science
Foundation. Both authors are supported by the Jump Trading Mathlab Research
Fund.

###### Abstract

The smooth (resp. metric and complex) Nielsen Realization Problem for K3
surfaces $M$ asks: when can a finite group $G$ of mapping classes of $M$ be
realized by a finite group of diffeomorphisms (resp. isometries of a Ricci-
flat metric, or automorphisms of a complex structure)? We solve the metric and
complex versions of Nielsen Realization, and we solve the smooth version for
involutions. Unlike the case of $2$-manifolds, some $G$ are realizable and
some are not, and the answer depends on the category of structure preserved.
In particular, Dehn twists are not realizable by finite order diffeomorphisms.
We introduce a computable invariant $\mathbf{L}_{G}$ that determines in many
cases whether $G$ is realizable or not, and apply this invariant to construct
an $S_{4}$ action by isometries of some Ricci-flat metric on $M$ that
preserves no complex structure.

We also show that the subgroups of $\operatorname{Diff}(M)$ of a given prime
order $p$ which fix pointwise some positive-definite $3$-plane in
$H_{2}(M;\mathbb{R})$ and preserve some complex structure on $M$ form a single
conjugacy class in $\operatorname{Diff}(M)$ (it is known that then
$p\in\\{2,3,5,7\\}$).

###### Contents

  1. 1 Introduction
    1. 1.1 A computable invariant
    2. 1.2 Metric and complex Nielsen Realization
    3. 1.3 (Non)realization for cyclic subgroups
    4. 1.4 Concluding remarks
  2. 2 Metric and complex Nielsen Realization
    1. 2.1 The Torelli Theorem for K3 surfaces
    2. 2.2 Completing the proofs of metric and complex Nielsen realization
  3. 3 The fixed set of a diffeomorphism of prime order
    1. 3.1 The Hirzebruch $G$-signature theorem
    2. 3.2 $H_{2}(N;\mathbb{Z})$ as a $\mathbb{Z}[G]$-module
    3. 3.3 Cyclotomic summands and topology of the fixed set
    4. 3.4 Smooth involutions have finite fixed set
    5. 3.5 Resolution of a spin orbifold of dimension 4
  4. 4 Prime order subgroups of $\operatorname{Diff}(M)$
    1. 4.1 A first reduction for the case $p$ odd
    2. 4.2 The resolved orbifold $M_{G}$
    3. 4.3 The Nikulin lattice
    4. 4.4 A Leech type sublattice of the Nikulin lattice
    5. 4.5 The case $p=2$
    6. 4.6 Nielsen realization relative to a Ricci-flat metric
    7. 4.7 Elliptic fibrations

##  1 Introduction

Let $M$ be a closed, oriented, smooth manifold. The (smooth) mapping class
group of $M$ is the group
$\operatorname{Mod}(M):=\pi_{0}(\operatorname{Diff}^{+}(M))$ of smooth isotopy
classes of orientation-preserving diffeomorphisms of $M$. The Nielsen
Realization Problem asks if the natural map
$\operatorname{Diff}^{+}(M)\to\operatorname{Mod}(M)$ has a section over any
finite subgroup $G\subset\operatorname{Mod}(M)$; that is, does any finite
group of mapping classes arise as a finite group of diffeomorphisms? When
there is such a section we say that $G$ is realized as a group of
diffeomorphisms. Whenever $M$ admits a particular kind of structure, there is
a corresponding Nielsen realization problem. For example, one can also ask:

Metric Nielsen Realization:  Suppose that $M$ admits a distinguished class
$\mathcal{C}$ of metrics. Given a finite subgroup
$G\subset\operatorname{Mod}(M)$, does there exist a lift of $G$ to
$\operatorname{Diff}(M)$, and a metric $g\in\mathcal{C}$, such that
$G\subseteq{\rm Isom}(M,g)$?

Complex Nielsen Realization:  Given a finite subgroup
$G\subset\operatorname{Mod}(M)$, does there exist a lift of $G$ to
$\operatorname{Diff}(M)$, and a complex structure $J$ on $M$, such that
$G\subseteq\operatorname{Aut}(M,J)$?

For $M$ a closed, oriented surface $\Sigma_{g}$, a positive answer in all
categories was given by Fenchel for cyclic $G$ and by Kerckhoff in general:
any finite subgroup $G\subset\operatorname{Mod}(\Sigma_{g})$ is realized by a
group of automorphisms of both a complex structure on $\Sigma_{g}$ and its
associated constant curvature metric.

In this paper we consider the Nielsen Realization Problem for K3 surfaces $M$.
These admit both complex structures and Ricci-flat metrics. We solve the
metric and complex Nielsen Realization (Theorem 1.2) problems; we solve the
smooth version for involutions (Theorem 1.8); and we give partial progress in
general. Unlike the case of $2$-manifolds, some $G$ are realizable and some
are not, and the answer depends on the category of structure preserved. We
introduce a computable invariant $\mathbf{L}_{G}$ that determines in many
cases if $G$ is realizable or not. Two other results are: the classification
of prime order subgroups of $\operatorname{Diff}(M)$ that preserve some
complex structure and pointwise fix a positive definite $3$-plane in
$H_{2}(M;\mathbb{R})$; and a characterization of “Nikulin type” involutions in
$\operatorname{Diff}(M)$.

###  1.1 A computable invariant

Recall that a K3 surface is a closed, simply-connected complex surface that
admits a nowhere vanishing holomorphic $2$-form. It is known that any two K3
surfaces are diffeomorphic (see for example [BPHV], Cor. 8.6); we call this
diffeomorphism type the K3 manifold. We fix one such $M$ throughout this
paper. It is also true that, as Friedman-Morgan have shown ([FM], Ch. VII Cor.
3.5), any complex structure on $M$ turns it into a K3 surface. The manifold
$M$ has a natural fundamental class $[M]\in H_{4}(M;\mathbb{Z})$: it is the
one for which the signature of $M$ is $-16$. So every diffeomorphism of $M$ is
necessarily orientation-preserving.

It is known that $H_{2}(M):=H_{2}(M;\mathbb{Z})\cong\mathbb{Z}^{22}$, and that
$H_{2}(M)$ as a lattice equipped with the symmetric, bilinear intersection
pairing (with respect to the above-mentioned orientation)

$H_{2}(M)\times H_{2}(M)\to\mathbb{Z},\quad(a,b)\mapsto a\cdot b$

is isomorphic to the K3 lattice $\mathbf{U}^{\oplus
3}\oplus\mathbf{E}_{8}(-1)^{\oplus 2}$, where $\mathbf{U}$ denotes the
hyperbolic lattice (which has two isotropic generators $e,f$ with $e\cdot
f=1$), the term $\mathbf{E}_{8}(-1)$ denotes the $\mathbf{E}_{8}$-lattice
whose pairing has been multiplied with $-1$, and $\oplus$ stands for the
orthogonal direct sum. In particular $H_{2}(M)$ has signature type $(3,19)$.
We will often write $\mathbf{\Lambda}$ for the lattice $H_{2}(M)$ and
$\mathbf{\Lambda}_{R}$ for $\mathbf{\Lambda}\otimes R$ when $R$ is an abelian
group. Note that the group $\operatorname{O}(\mathbf{\Lambda})$ of
automorphisms of $\mathbf{\Lambda}$ is an arithmetic lattice in the real
semisimple Lie group $\operatorname{O}(\mathbf{\Lambda}_{\mathbb{R}})$.

The natural action of $\operatorname{Diff}(M)$ on $\mathbf{\Lambda}$ factors
through an orthogonal representation
$\rho:\operatorname{Mod}(M)\to\operatorname{O}(\mathbf{\Lambda})$. According
to Donaldson ([D-or] (3.20) and Definition (3.24)), the tautological $3$-plane
bundle over the space $Gr^{+}_{3}(\mathbf{\Lambda}_{\mathbb{R}})$ of positive-
definite $3$-planes in $\mathbf{\Lambda}_{\mathbb{R}}$ comes with a natural
orientation (which we shall call a _spinor orientation_), so that $\rho$ will
take its values in the subgroup of $g\in\operatorname{O}(\mathbf{\Lambda})$
that preserve this orientation. This subgroup is of index two in
$\operatorname{O}(\mathbf{\Lambda})$ and will be denoted by
$\operatorname{O}^{+}(\mathbf{\Lambda})$. For example, if
$v\in\mathbf{\Lambda}$ is such that $v\cdot v=-2$ (we will refer to such a $v$
as a _$(-2)$ -vector_ in $\mathbf{\Lambda}$), then the orthogonal reflection
with respect to $v$ is given by $x\in\mathbf{\Lambda}\mapsto x+(v\cdot x)v$
and lies in $\operatorname{O}^{+}(\mathbf{\Lambda})$. Such a reflection is the
image under $\rho$ of a “$2$-dimensional Dehn twist” about a $2$-sphere in $M$
(see [Se] for a precise definition of these). Since these reflections generate
all of $\operatorname{O}^{+}(\mathbf{\Lambda})$, the representation $\rho$ has
image $\operatorname{O}^{+}(\mathbf{\Lambda})$ (see [Bo] and [Ma1]).

The theory of complex K3 surfaces produces many finite subgroups
$G\subset\operatorname{Mod}(G)$ that are realizable by complex (even Kähler)
automorphisms. The question is whether or not these are the only realizable
finite subgroups $G\subset\operatorname{Mod}(M)$. We now introduce an
invariant to help answer this question.

Let $G\subset\operatorname{O}^{+}(\mathbf{\Lambda})$ be a finite subgroup.
Since $Gr_{3}^{+}(\mathbf{\Lambda}_{\mathbb{R}})$ is the symmetric space of
$\operatorname{O}^{+}(\mathbf{\Lambda}_{\mathbb{R}})$ it is nonpositively
curved, and so $G$ will have a fixed point in this space. In other words, $G$
will leave invariant a positive definite $3$-plane
$P\subset\mathbf{\Lambda}_{\mathbb{R}}$. Since $G$ also preserves the
orientation of $P$, the representation $\rho$ restricts to a representation
$\rho_{P}:G\to\operatorname{SO}(P)$. The image $\rho_{P}(G)$ will be conjugate
to one of the familiar groups: cyclic (this includes the trivial group),
dihedral, tetrahedral, octahedral or icosahedral. Note that only the last
three cases yield an irreducible representation.

Let $\mathbf{I}_{G}\subset\mathbf{\Lambda}_{\mathbb{R}}$ be the sum of the
irreducible $\mathbb{R}[G]$-submodules of the type that appear in $P$. As our
notation intends to suggest, this subspace (in contrast to $P$) is canonically
associated with $G$.

######  Definition 1.1 (The invariant $\mathbf{L}_{G}$).

Define $\mathbf{L}_{G}$ to be the following sublattice of $\mathbf{\Lambda}$:

$\mathbf{L}_{G}:=\mathbf{I}_{G}^{\perp}\cap\mathbf{\Lambda}.$

The sublattice $\mathbf{L}_{G}$ is clearly negative definite and
$G$-invariant. It is easily computable.

###  1.2 Metric and complex Nielsen Realization

Siu [Siu] proved that every complex K3 surface $M$ admits a Kähler metric and
Yau [Yau] proved that for every Kähler metric on $M$ there is a unique Ricci-
flat Kähler metric in its cohomology class. These Ricci-flat metrics play the
same role for K3 surfaces that constant curvature metrics play for Riemann
surfaces. If a finite group $G\subset\operatorname{Diff}(M)$ preserves some
complex structure on $M$ then one can take a $G$-average of a Kähler metric
and then apply Yau’s theorem to find a $G$-invariant, Ricci-flat Kähler
metric. Thus there are basically three Nielsen Realization problems for K3
surfaces, each stronger than the previous: realization by diffeomorphisms of
$M$; realization by isometries of some Ricci-flat metric on $M$; and
realization by complex automorphism of $M$.

For the complex and Ricci-flat categories there is an associated Teichmüller
space, which is particularly nice in the Ricci-flat case. For our purposes the
latter case is the most relevant. Let $\operatorname{Teich}(M)$ denote the
moduli space of smooth isotopy classes of unit volume, Ricci-flat metrics on
$M$; thus two such metrics define the same point of $\operatorname{Teich}(M)$
if one is the pullback of the other via a diffeomorphism of $M$ isotopic to
the identity. The space $\operatorname{Teich}(M)$ comes equipped with a
natural $\operatorname{Mod}(M)$-action. It also comes with a period map: for a
(Ricci-flat) Riemann metric on $M$, its space of self-dual harmonic $2$-forms
defines a positive definite $3$-plane in $H^{2}(M;\mathbb{R})$; its orthogonal
complement is negative definite and is defined by the space anti-self-dual
harmonic $2$-forms. Identifying $H^{2}(M;\mathbb{R})$ with
$\mathbf{\Lambda}_{\mathbb{R}}$ via Poincaré duality, this gives a map

$\mathscr{P}:\operatorname{Teich}(M)\to
Gr^{+}_{3}(\mathbf{\Lambda}_{\mathbb{R}}).$

Since the codomain of $\mathscr{P}$ is the symmetric space for
$\operatorname{O}(\mathbf{\Lambda}_{\mathbb{R}})$, the group
$\operatorname{O}^{+}(\mathbf{\Lambda})$ acts properly discontinuously on it.

We will need some known properties of $\mathscr{P}$ which we recall in more
detail in Proposition 2.1. The image of $\mathscr{P}$ is the subset $U\subset
Gr^{+}_{3}(\mathbf{\Lambda}_{\mathbb{R}})$ given by the set of all positive-
definite $3$-planes not perpendicular to a $(-2)$-vector. This $U$ is a
simply-connected open subset of $Gr^{+}_{3}(\mathbf{\Lambda}_{\mathbb{R}})$.
The natural action of $\operatorname{Diff}(M)$ on $\mathbf{\Lambda}$ factors
through a representation
$\rho:\operatorname{Mod}(M)\to\operatorname{O}^{+}(\mathbf{\Lambda})$ whose
kernel $\operatorname{{\mathcal{I}}}(M):=\ker(\rho)$ is called the Torelli
subgroup of $\operatorname{Mod}(M)$. The map $\mathscr{P}$ is obviously
$\operatorname{Mod}(M)$-equivariant, with $\operatorname{Mod}(M)$ acting on
$Gr^{+}_{3}(\mathbf{\Lambda}_{\mathbb{R}})$ via $\rho$. It maps every
connected component of $\operatorname{Teich}(M)$ homeomorphically onto $U$,
thus making $\operatorname{Teich}(M)$ a smooth manifold of dimension $3\cdot
19=57$. If $\operatorname{Teich}(M)_{o}$ is a connected component of
$\operatorname{Teich}(M)$, then the $\operatorname{Mod}(M)$-stabilizer
$\operatorname{Mod}(M)_{o}$ of $\operatorname{Teich}(M)_{o}$ maps
isomorphically onto $\operatorname{O}^{+}(\mathbf{\Lambda})$. Thus
$\operatorname{Mod}(M)$ acquires the structure of a semi-direct product

$\operatorname{Mod}(M)=\operatorname{{\mathcal{I}}}(M)\rtimes\operatorname{Mod}(M)_{o}\cong\operatorname{{\mathcal{I}}}(M)\rtimes\operatorname{O}^{+}(\mathbf{\Lambda})$

and gives a corresponding description of $\operatorname{Teich}(M)$ as a
$\operatorname{Mod}(M)$-manifold:

$\operatorname{Teich}(M)\cong\operatorname{Mod}(M)\times_{\operatorname{Mod}(M)_{o}}U.$

In particular, $\operatorname{{\mathcal{I}}}(M)$ permutes simply-transitively
the connected components of $\operatorname{Teich}(M)$.

A finite subgroup of $\operatorname{Diff}(M)$ that fixes some Ricci-flat
metric on $M$ always fixes a point of $\operatorname{Teich}(M)$ and hence a
connected component of $\operatorname{Teich}(M)$. In view of the preceding it
is then natural to consider the metric Nielsen Realization problem for finite
subgroups of $\operatorname{O}^{+}(\mathbf{\Lambda})$, although this is
clearly _a priori_ not the same as the corresponding problem for finite
subgroups of $\operatorname{Mod}(M)$. The following theorem solves this
version of the Ricci-flat and complex versions of the Nielsen Realization
problem for K3 surfaces. The answer to the homological version is in terms of
the invariant $\mathbf{L}_{G}$ only.

######  Theorem 1.2 (Metric and complex Nielsen Realization).

Let $M$ be a K3 manifold and let $G$ be a finite subgroup of
$\operatorname{O}^{+}(\mathbf{\Lambda})$.

  1. 1.

$G$ lifts to the group of isometries of a Ricci-flat metric on $M$ if and only
if $\mathbf{L}_{G}$ contains no $(-2)$-vectors.

  2. 2.

$G$ lifts to the group of automorphisms of some complex structure on $M$
endowed with a Ricci-flat Kähler metric (making it a _Kähler-Einstein_ K3
surface) if and only if $\mathbf{L}_{G}$ contains no $(-2)$-vectors and in
addition the trivial representation appears in $\mathbf{L}_{G}^{\perp}$. In
this case the complex structure can be chosen so that $M$ is projective and
$G$ acts by algebraic automorphisms.

If instead $G$ is given rather as a finite subgroup of
$\operatorname{Mod}(M)$, then a necessary and sufficient condition for it to
lift as in Case 1 (resp. Case 2) is that $G$ satisfies the homological
conditions stated in that case and also preserves a connected component of the
Teichmüller space $\operatorname{Teich}(M)$.

######  Remark 1.3.

We do not know whether conversely, a finite subgroup
$G\subset\operatorname{Diff}(M)$ necessarily preserves a connected component
of $\operatorname{Teich}(M)$, or equivalently (by Theorem 1.2 above) a Ricci-
flat metric. Perhaps this might be proved by choosing a $G$-invariant metric
and then use the Ricci flow to deform it equivariantly into a $G$-invariant
Ricci-flat metric. (As Shing-Tung Yau pointed out to us, it suffices that one
ends up with a metric of nonnegative scalar curvature, since for a K3 manifold
this implies Ricci-flatness.)

The following result shows that the metric and complex versions of Nielsen
Realization are different.

######  Theorem 1.4 (Ricci-flat but not complex).

There exists a subgroup $G\subset\operatorname{O}^{+}(\mathbf{\Lambda})$
isomorphic to the alternating group $\mathcal{A}_{4}$ that can be realized as
a group of isometries of some Ricci-flat metric on $M$, but that preserves no
complex structure on $M$.

The proof of Theorem 1.4 is an example of both a constructive and an
obstructive application of Theorem 1.2: the latter result implies that, to
find the required example, it suffices to find a subgroup
$G\subset\operatorname{O}^{+}(\mathbf{\Lambda})$ such that $\mathbf{L}_{G}$
contains no $(-2)$-vector and $\mathbf{L}_{G}^{\perp}$ contains no $G$-fixed
vector. This is a linear algebra problem.

Prime order subgroups. When $G$ pointwise fixes a positive-definite $3$-plane,
it will also fix a vector in $\mathbf{\Lambda}$ with positive self-
intersection, and then both parts of Theorem 1.2 can be refined: an invariant
Ricci-flat metric is always accompanied by an invariant complex structure for
which this is a Kähler metric. We also note that Theorem 1.2 implies the
following.

######  Corollary 1.5.

A cyclic subgroup $G\subset\operatorname{Diff}(M)$ of prime order that leaves
invariant a connected component of $\operatorname{Teich}(M)$ and a positive-
definite $3$-plane $P\subset\mathbf{\Lambda}_{\mathbb{R}}$ but acts
_nontrivially_ on $P$, preserves a Ricci-flat metric.

Indeed, in this case $\mathbf{L}_{G}=\\{0\\}$ and so the assumption of part
(1) is trivially satisfied. For this reason we will mostly focus on the case
when $G$ fixes a positive-definite $3$-plane in
$\mathbf{\Lambda}_{\mathbb{R}}$ pointwise.(111In the literature, this
condition is often called symplectic.) Here part (2) of Theorem 1.2 can be
both refined and be made more concrete. We first state a general result, and
then indicate how these are realized on elliptically fibered K3 surfaces. We
should add however, that this is essentially hidden in the K3 literature and
so here our main contribution is to bring this to light and to present this in
a uniform manner.

######  Theorem 1.6 (Single conjugacy class).

Let $M$ be a K3 manifold. Let $G\subset\operatorname{Diff}(M)$ have prime
order $p$ and fix pointwise a positive-definite $3$-plane in
$\mathbf{\Lambda}_{\mathbb{R}}$ and a complex structure on $M$ (or
equivalently, a Ricci-flat metric). Then $p\in\\{2,3,5,7\\}$ and for each such
$p$, the subgroups of $\operatorname{Diff}(M)$ as above make up a single
conjugacy class in $\operatorname{Diff}(M)$. Each such $G$ can be realized
geometrically inside the Mordell-Weil group of an elliptic fibration (see
below).

To elaborate on the last point (and as will be described in detail in
Subsection 4.7), for each $(p,\nu)\in\\{(2,8),(3,6),(5,4),(7,3)\\}$ there
exists a complex K3 surface $X$, elliptically fibered with $\nu$ fibers of
Kodaira type $I_{p}$ and $\nu$ fibers of type $I_{1}$ (222Recall that an
$I_{m}$-fiber is for $m>1$ a ‘cycle’ of $(-2)$-curves, $m$ in number, and an
$I_{1}$ fiber is a rational curve with a node.), admitting an automorphism of
order $p$; see Figure 1. This automorphism is then realized inside its
Mordell-Weil group (i.e. the group of automorphisms that preserve each fiber
and act on it as a translation): it acts in each $I_{p}$-fiber as a nontrivial
‘rotation’ (but not necessarily over the same angle - see below). It also acts
nontrivially in each $I_{1}$-fiber by “rotation” of the rational curve, fixing
its unique singular point. See Figure 1. This accounts for all the $G$-fixed
points on $M$.

The proof of Theorem 1.6, which as we noted above is implicit in the
literature, involves a substantial amount of nontrivial lattice theory. We
will treat this aspect in a way that is both more conceptual and less
computational than in the sources of which we are aware (so that this part
might also be of some use for readers mostly interested in K3 surfaces). Along
the way we find a construction of the Coxeter-Todd lattice that is possibly
new.

Figure 1: The elliptically fibered K3 surface $X_{5}$. The generic fiber is a
smooth elliptic curve. There are $4$ fibers of type $I_{5}$, consisting of
length $5$ $\mathbb{P}^{1}$-chains, and $4$ fibers that are rational nodal
curves. The cyclic group of order $5$ acts on $X_{5}$ as the Mordell-Weil
group of $X_{5}$; that is, the group of complex automorphisms of $X_{5}$ that
preserve each fiber, translate each smooth fiber, rotate the $I_{5}$ fibers,
and rotate the rational nodal curves fixing the singularity. The rotation
angles on two of the $I_{5}$ (resp. $I_{1}$) fibers are twice that of the
other two (cf. Remark 4.6).

######  Remark 1.7.

The complex structures on $M$ given up to a diffeomorphism that acts trivially
on $H_{2}(M)$ are parametrized by a connected, complex (but non-Hausdorff)
manifold of complex dimension $20$. Those that admit a group of complex
automorphisms of order $p$ make up a locus of complex dimension $2\nu-4$
($=12,8,4,2$ when $p=2,3,5,7$). Theorem 1.6 then amounts to the statement that
the irreducible components of this locus are transitively permuted by
$\operatorname{O}^{+}(H_{2}(M))$. Those that have an elliptic fibration make
up a locus of complex codimension $1$ (so of dimension $2\nu-5$), and those
that have in addition a section a locus of complex codimension $2$ (and so of
dimension $2\nu-6$; note this is then a finite set when $p=7$).

In other words, there are families of prime order $G$ acting by complex
automorphisms on K3s admitting no elliptic fibration (in fact this is generic
among K3s with a $G$-action), and yet Theorem 1.6 gives for each $p$ that any
such action is smoothly conjugate to the specific elliptically fibered
“translation” action given above (cf. Figure 1).

###  1.3 (Non)realization for cyclic subgroups

Even in the smooth category, and even for involutions, there are obstructions
for $G\subset\operatorname{Mod}(M)$ to be realized.

######  Theorem 1.8 (Smooth Nielsen (non)realization for involutions).

Let $G\subset\operatorname{Mod}(M)$ be cyclic of order $2$. Then the
nontrivial element of $G$ is realized by a smooth involution of $M$ if and
only if $\mathbf{L}_{G}$ contains no $(-2)$-vectors. In this case the
involution preserves some Ricci-flat Kähler structure on $M$.

######  Remark 1.9.

Note that Theorem 1.2 gives the ‘if’ part of the last sentence of Theorem 1.8.
The main content of Theorem 1.8 is therefore the ‘only if’ part.

An interesting example to which Theorem 1.8 applies is a Dehn twist
$T_{S}:M\to M$ about an embedded $2$-sphere $S\subset M$ with $S\cdot S=-2$
(this intersection number is independent of the orientation of $S$). It is
known (see, e.g.[Se], Proposition 1.1) that $T_{S}^{2}$ is smoothly isotopic
to the identity. However, Theorem 1.8 (or its proof, for the stronger result
claimed) implies the following (which we will prove in detail below.)

######  Corollary 1.10 (Twists not realizable).

Let $T_{S}\in\operatorname{Diff}(M)$ be the Dehn twist about an embedded
$2$-sphere $S\subset M$ with $S\cdot S=-2$. Although $T_{S}^{2}$ is smoothly
isotopic to the identity, $T_{S}$ is not even topologically isotopic to any
finite order diffeomorphism of $M$.

One can also show for example that products of disjoint twists cannot be
smoothly realized as finite subgroups of $\operatorname{Diff}(M)$.

Involutions of Nikulin type. Nikulin showed that the involutions of a
$K3$-surface $X$ that act as the identity on $H^{2,0}(X)$ are all alike: they
have eight fixed points, so that the orbit space will have eight-double
points, and resolving these double points produces a $K3$-surface. The K3
surfaces endowed with such an involution make up a single connected family. As
Morrison [Mor] observed, the action of the involution on their second homology
is equivalent to the involution of the $K3$-lattice $\mathbf{U}^{\oplus
3}\oplus\mathbf{E}_{8}(-1)^{\oplus 2}$ that exchanges the last two summands.
This motivates the following definition.

######  Definition 1.11.

An element of order $2$ in $\operatorname{Diff}(M)$ is _topologically_ resp.
_differentiably of Nikulin type_ if $f$ has eight fixed points points and the
orbit space is, after a simple resolution of the resulting $8$ quotient
singularities, a closed $4$-manifold homeomorphic resp. diffeomorphic to a
$K3$-surface.

The main ingredient in our proof of Theorem 1.8 is the following theorem,
which characterizes involutions of Nikulin type in terms of purely homological
data. We remind the reader that any finite order diffeomorphism leaves
invariant a positive $3$-plane in $\mathbf{\Lambda}_{\mathbb{R}}$.

######  Theorem 1.12 (Involutions of Nikulin type).

A smooth involution of the $K3$ manifold $M$ that fixes pointwise some
positive $3$-plane in $\mathbf{\Lambda}_{\mathbb{R}}$ is topologically of
Nikulin type and acts on $\mathbf{\Lambda}$ as a Nikulin involution.

Cyclic subgroups of order $p\geq 3$. Unlike for the $p=2$ case (Theorem 1.8)
we do not have a definitive result for $p$ odd, but we are able to prove in
many cases a dichotomy for realizable cyclic $G$ of prime order, into “Nikulin
type” and “Coxeter type”; see Theorem 4.1.

###  1.4 Concluding remarks

We state here a basic result regarding the finite subgroups of
$\operatorname{O}^{+}(\mathbf{\Lambda})$, which although not used in this
paper, might be helpful for what follows. Let $G$ be a finite subgroup of
$\operatorname{O}^{+}(\mathbf{\Lambda})$. The reflections with respect to
$(-2)$-vectors in the negative definite lattice $\mathbf{L}_{G}$ generate a
finite Coxeter subgroup $W\subset\operatorname{O}^{+}(\mathbf{\Lambda})$. If
$C$ is a connected component of the complement of the union of the reflection
hyperplanes in $\mathbf{L}_{G}\otimes\mathbb{R}$ with respect to the
$(-2)$-reflections in $W$ (a _chamber_), then $G=G_{C}\rtimes(W\cap G)$, where
$G_{C}$ denotes the $G$-stabilizer of $C$. The Torelli theorem for K3 surfaces
implies that there exists a complex structure $J$ on $M$ such that $G_{C}$ is
realized as a group of automorphisms of $(M,J)$. Although Theorem 1.10 shows
that any $(-2)$-reflection in $W$ cannot be realized as a smooth involution,
it is possible for $W\cap G$ to be nontrivial without containing such elements
and so we cannot exclude the possibility that $G$ lifts to
$\operatorname{Diff}(M)$, even when $W\cap G\not=\\{1\\}$.

Methods. The methods used in this paper are varied. To constrain fixed sets we
use the theory of finite group actions on manifolds (e.g. the Hirzebruch
$G$-signature theorem and Smith theory by way of the work of Edmonds), and an
adjunction formula of Kronheimer-Mrowka coming out of Seiberg-Witten theory.
We analyze $\operatorname{Mod}(M)$ via its action on an appropriate
Teichmüller space, using a differential-geometric version of the Torelli
theorem for K3 surfaces. Finally, much of the work in this paper involves
lattice theory.

Relation to other work. Giansiracusa-Kupers-Tshishiku [GKT] proved that the
surjection $\pi:\operatorname{Diff}(M)\to\operatorname{Mod}(M)$ does not split
over any finite index subgroup $\Gamma\subseteq\operatorname{Mod}(M)$.
Corollary 1.10 implies this theorem when $\Gamma=\operatorname{Mod}(M)$;
indeed it gives that $\pi$ does not even split over certain order $2$
subgroups of $\operatorname{Mod}(M)$. We will see below, however, that this
corollary can also be derived from a theorem of Bryan [Bry]. We also mention
that Baraglia and Konno [BK] recently constructed a topological involution of
$M$ which is not conjugate to a differentiable involution.

Since all finite group actions on a K3 manifold that we are aware of fix a
Ricci-flat metric we are led to ask:

######  Question 1.13.

Does every finite subgroup of $\operatorname{Diff}(M)$ leave invariant a
Ricci-flat metric?

Another “question” (which we do not wish to advertise as explicitly as the one
above, because it is too vague) is the following. As we indicated, the finite
subgroups of $\operatorname{Diff}(M)$ that fix a positive-definite $3$-plane
in $\mathbf{\Lambda}_{\mathbb{R}}$ pointwise correspond to what in the K3
community are called symplectic automorphism groups. A well-known theorem of
Mukai [Mu] characterizes the isomorphism classes of such groups as the
subgroups of the Mathieu group $M_{24}$ subject to certain properties
($M_{24}$ is a subgroup of the symmetric group $\mathcal{S}_{24}$ and the
subgroups $G\subset M_{24}$ in question are those that fix an element and have
at least $5$ orbits). Since a clear conceptual link between these two classes
of finite groups has not been found yet, one may wonder whether the passage to
$\operatorname{Mod}(M)$ or $\operatorname{Diff}(M)$ can shed some light on
this.

######  Question 1.14.

The subgroup of $\operatorname{Mod}(M)$ generated by the Dehn twists defined
by smoothly embedded $(-2)$-spheres in $M$ generate a normal subgroup
$\operatorname{Dehn}(M)\subset\operatorname{Mod}(M)$. It is not hard to see
that $\operatorname{Dehn}(M)$ contains the stabilizer of every connected
component of $\operatorname{Teich}(M)$. This leads us to ask: what is
$\operatorname{{\mathcal{I}}}(M)\cap\operatorname{Dehn}(M)$? Note that if that
intersection is trivial, then we have a direct product decomposition
$\operatorname{Mod}(M)=\operatorname{{\mathcal{I}}}(M)\times\operatorname{Dehn}(M)$
with $\rho$ mapping $\operatorname{Dehn}(M)$ isomorphically onto
$\operatorname{O}^{+}(\mathbf{\Lambda})$. But it could well be that
$\operatorname{Dehn}(M)$ is of finite index in $\operatorname{Mod}(M)$, if not
all of $\operatorname{Mod}(M)$ just as for closed, connected, oriented
2-manifolds.

Acknowledgements. We would like to thank Andras Stipsicz and Shing-Tung Yau
for helpful email exchanges, and David Baraglia, Trevor Hyde and Jim Davis for
comments and corrections on an earlier draft of this paper. We are grateful to
Curt McMullen for his extensive comments on that draft, and for his many
useful suggestions. Finally, we thank two anonymous referees for their
comments and corrections.

##  2 Metric and complex Nielsen Realization

The goal of this section is to prove Theorem 1.2 and Theorem 1.4.

Notation. By a _lattice_ we shall always mean a free abelian group
$\mathbf{L}$ of finite rank endowed with symmetric bilinear form
$(v,v^{\prime})\in\mathbf{L}\times\mathbf{L}\mapsto v\cdot
v^{\prime}\in\mathbb{Z}$. In this context a direct sum is always understood as
an orthogonal direct sum.

Unless mentioned otherwise, we will assume that a lattice $\mathbf{L}$ is
nondegenerate, so that the associated map from $\mathbf{L}$ to its dual
$\mathbf{L}^{\vee}:=\operatorname{Hom}(\mathbf{L},\mathbb{Z})$ is injective.
We then identify $\mathbf{L}^{\vee}$ with the set of
$x\in\mathbb{Q}\otimes\mathbf{L}$ for which $x\cdot v\in\mathbb{Z}$ for all
$v\in\mathbf{L}$. The _discriminant group_ $\mathbf{L}^{\vee}/\mathbf{L}$ is
finite and inherits from $\mathbf{L}$ a symmetric bilinear form that takes
values in $\mathbb{Q}/\mathbb{Z}$. In case the form is even, so that it comes
from a quadratic form (namely $v\in\mathbf{L}\to\frac{1}{2}v\cdot
v\in\mathbb{Z}$) and the discriminant group inherits such a form as well
(again taking values in $\mathbb{Q}/\mathbb{Z})$. If $n$ is a nonzero integer,
then we denote by $\mathbf{L}(n)$ the lattice whose underlying group is
$\mathbf{L}$, but for which the form has been multiplied by $n$. So
$\mathbf{L}(n)^{\vee}=\frac{1}{n}\mathbf{L}^{\vee}$.

We make the notational distinction between the type of a root system of equal
root length and the lattice it generates (for which all roots have square
length $2$) by in the latter case using a bold font. For example,
$\mathbf{E}_{8}$ is an even unimodular positive definite lattice of rank $8$
and its vectors $v$ with $v\cdot v=2$ make up a root system of type $E_{8}$.

###  2.1 The Torelli Theorem for K3 surfaces

Before we begin the proof of Theorem 1.2 proper, we review the various
formulations of the Torelli theorem for K3 surfaces (see for example [Lo]).
Suppose $M$ is endowed with a complex structure $J$ (so that by [FM] it
becomes a K3 surface). It is known that the automorphism group
$\operatorname{Aut}(M,J)$ acts faithfully on $\mathbf{\Lambda}$ and so can be
considered as a subgroup of $\operatorname{O}^{+}(\mathbf{\Lambda})$. The
complex structure gives rise to a Hodge decomposition of $\mathbf{\Lambda}$
with $\dim H^{2,0}(M,J)=1$, and basic Hodge theory tells us that this complex
line is isotropic for the complex-bilinear extension of the intersection
pairing and positive definite for the hermitian extension. This implies that
$H^{2,0}(M,J)$ maps under the projection
$\mathbf{\Lambda}_{\mathbb{C}}\to\mathbf{\Lambda}_{\mathbb{R}}$ (taking the
real part) onto a positive-definite $2$-plane
$\Pi_{J}\subset\mathbf{\Lambda}_{\mathbb{R}}$. This $2$-plane $\Pi_{J}$
inherits a natural orientation from $H^{2,0}(M,J)$, and as such it is easily
seen to be a complete invariant of the Hodge structure.

The Torelli theorem states that more is true: $\Pi_{J}$ determines $J$ up to a
diffeomorphism that fixes $\Pi_{J}$ pointwise. It also tells us that every
oriented, positive-definite $2$-plane $\Pi$ so arises. In other words, such a
$\Pi$ determines a complex structure $J$ up to diffeomorphism that fixes $\Pi$
pointwise. However that diffeomorphism may act nontrivially on
$\mathbf{\Lambda}$. Indeed, $\operatorname{Aut}(M,J)$ can be a proper subgroup
of the $\operatorname{O}^{+}(\mathbf{\Lambda})$-stabilizer
$\operatorname{O}^{+}(\mathbf{\Lambda})_{\Pi}$ of $\Pi$.

In order to state a more precise result, first note that the orthogonal
complement $\Pi^{\perp}\subset\mathbf{\Lambda}_{\mathbb{R}}$ has hyperbolic
signature $(1,19)$, so the set of $x\in\Pi^{\perp}$ with $x\cdot x>0$ consists
of a pair of antipodal cones. The spinor orientation and the given orientation
of $\Pi$ single out one of them, denoted $C_{\Pi}$, to which we will refer as
the _positive cone_. The subgroup
$W_{\Pi}\subset\operatorname{O}^{+}(\mathbf{\Lambda})_{\Pi}$ generated by the
orthogonal reflections in the $(-2)$-vectors in
$\mathbf{\Lambda}\cap\Pi^{\perp}$ preserves $C_{\Pi}$ and acts on it as a
Coxeter group. The collection of reflection hyperplanes is locally finite on
$C_{\Pi}$ and decomposes $C_{\Pi}$ into faces. The group $W_{\Pi}$ acts simply
transitively of the set of open faces. We refer to these as _$\Pi$ -chambers_.

One way to phrase a strong form of the Torelli theorem is the assertion that
for every $\Pi$-chamber $K\subset C_{\Pi}$ there exists a complex structure
$J$ for which $\Pi_{J}=\Pi$ and $K$ is the set of classes of Kähler forms
relative to $J$. This complex structure is indeed unique up to a
diffeomorphism that acts trivially on $\mathbf{\Lambda}$. Moreover,
$\operatorname{Aut}(M,J)$ maps isomorphically onto the
$\operatorname{O}^{+}(\mathbf{\Lambda})$-stabilizer of the pair $(\Pi,K)$. In
particular, to give a finite subgroup of $\operatorname{Aut}(M,J)$ is to give
a finite subgroup $G$ of $\operatorname{O}^{+}(\mathbf{\Lambda})_{\Pi,K}$.
Since $K$ is a convex (open) cone, this is equivalent to asking that $G$ has
fixed point $\kappa\in K$, which is then the class of a Kähler form.

There is a further refinement of the above that involves the twistor
construction and that is relevant here. According to Yau’s solution of the
Calabi conjecture, $\kappa$ is representable by a unique Kähler form for which
the associated metric $s$ is Ricci-flat (so that we have a Ricci-flat Kähler
metric). As a consequence, $G$ will fix this Ricci-flat Kähler metric. The
harmonic $2$-forms that are self-dual for the $\star$-operator define the
$3$-dimensional positive-definite space $P=\Pi+\mathbb{R}\kappa$. Note that by
construction, $P$ is not perpendicular to any $(-2)$-vector. The twistor
construction (see for example [HKLR]) shows that the
$\operatorname{SO}(P)$-orbit of the pair $(\Pi,\kappa)$ defines a family of
Ricci-flat Kähler structures on $M$ for which $s$ is the associated metric.
This enables us to state the Torelli theorem in differential-geometric terms:
For the purposes of reference we formulate this as a proposition.

######  Proposition 2.1 (The $3$-plane of self-dual harmonic forms).

Given any Ricci-flat metric $s$ on $M$ for which $M$ has unit volume, let
$P_{s}$ be the positive-definite $3$-plane of harmonic (in the $s$ metric)
self-dual $2$-forms. Then :

  1. 1.

$P_{s}$ is not perpendicular to any $(-2)$-vector.

  2. 2.

$P_{s}$ determines $s$ on $M$ up to an element of the Torelli group.

  3. 3.

Any positive-definite $3$-plane $P\subset\mathbf{\Lambda}_{\mathbb{R}}$ that
is not perpendicular to a $(-2)$-vector so arises.

For a proof of Proposition 2.1, see for example [Lo]. The following is a
corollary of the twistor construction.

######  Corollary 2.2.

Let $s$ be a Ricci-flat metric on $M$ and let
$P_{s}\subset\mathbf{\Lambda}_{\mathbb{R}}$ be the positive definite $3$-plane
defined by its harmonic self-dual $2$-forms. Then the isometry group of
$(M,s)$ is finite and its representation on $\mathbf{\Lambda}$ identifies this
group with the $\operatorname{O}^{+}(\mathbf{\Lambda})$-stabilizer of $P_{s}$.

###### Proof.

Suppose $g\in\operatorname{O}^{+}(\mathbf{\Lambda})$ leaves $P_{s}$ invariant.
Then $g$ must have finite order. It acts on $P_{s}$ as an element of
$\operatorname{SO}(P_{s})$ and so $P_{s}^{g}\not=\\{0\\}$. For each $\kappa\in
P_{s}^{g}\smallsetminus\\{0\\}$ the natural orientation on $P_{s}$ induces a
natural orientation on $\Pi_{\kappa}:=\kappa^{\perp}\cap P_{s}$. By the
twistor construction, $\Pi_{\kappa}$ defines a unique complex structure
$J_{\kappa}$ on $M$ for which $\Pi_{J_{\kappa}}=\Pi_{\kappa}$ as oriented
planes and for which $s$ is a Kähler metric.

The Torelli theorem then asserts that $g$ has a unique representative
$g_{\kappa}\in\operatorname{Diff}(M)$ that preserves both $J_{\kappa}$ and $s$
(or equivalently, $\kappa$ and $s$). Since the isometry group of $(M,s)$ is
discrete, $g_{\kappa}$ is independent of the connected component of $\kappa$
in $P_{s}^{g}\smallsetminus\\{0\\}$. If $P_{s}^{g}\smallsetminus\\{0\\}$ is
not connected then it is a punctured line and $\kappa\mapsto-\kappa$ exchanges
its components. This has the effect of changing the orientation of
$\Pi_{\kappa}$ and hence replaces the complex structure $J_{\kappa}$ by its
conjugate $\bar{J}_{\kappa}$. This will not affect $g_{\kappa}$ (for
$g_{\kappa}$ also preserves $\bar{J}_{\kappa}$) and so there is a unique lift
$\tilde{g}$ of $g$ to an isometry of $(M,s)$. Hence $g\mapsto\tilde{g}$
defines a group homomorphism that is a section of the map
$\operatorname{Aut}(M,s)\to\operatorname{O}^{+}(\mathbf{\Lambda})_{[P_{s}]}$.
The uniqueness assertion implies that the latter is injective. ∎

###  2.2 Completing the proofs of metric and complex Nielsen realization

With the above in hand we can complete the proof of Theorem 1.2.

###### Proof of Theorem 1.2.

We prove the nontrivial directions only (in the mapping class version we
assume that $G$ fixes a connected component of $\operatorname{Teich}(M)$). It
then remains to prove that if $\mathbf{L}_{G}$ contains no $(-2)$-vectors then
$G$ is realized as a group of isometries of a Ricci-flat metric. The other
properties already follow from our discussion above.

In the Grassmannian of $G$-invariant $3$-planes
$P\subset\mathbf{L}_{G}^{\perp}\otimes\mathbb{R}$, those with the property
that the rational hull of $P$ is $\mathbf{L}_{G}^{\perp}\otimes\mathbb{R}$ (or
equivalently, for which $P^{\perp}\cap\mathbf{\Lambda}=\mathbf{L}_{G}$) are
dense. Hence we can find a $G$-invariant, positive-definite $3$-plane
$P\subset\mathbf{L}^{\perp}_{G}\otimes\mathbb{R}$ for which
$P^{\perp}\cap\mathbf{\Lambda}=\mathbf{L}_{G}$. In that case $P^{\perp}$
contains no $(-2)$-vectors. By the differential-geometric version of the
Torelli theorem, there exists then a Ricci-flat metric $s$ on $M$ which in the
mapping class version can be taken to be in the component of
$\operatorname{Teich}(M)$ fixed by $G$ such that $P$ is the space defined by
the self-dual harmonic $2$-forms. Corollary 2.2 implies that $G$ then lifts to
a group of isometries of $(M,s)$. ∎

If $G$ acts irreducibly on $P$, and if $P^{\perp}$ does not contain any
$(-2)$-vector, then $G$ preserves a Ricci-flat metric without necessarily
preserving a complex structure on $M$. This explains why we put the emphasis
on the differential-geometric version of the Torelli theorem. Theorem 1.4,
which we now prove, gives an example that shows that this possibility actually
occurs.

###### Proof of Theorem 1.4.

We think of the symmetric group $\mathcal{S}_{4}$ as the Weyl group of a root
system $R$ of type $A_{3}$. It preserves the root lattice $\mathbf{A}_{3}$, a
free abelian group of rank $3$. $\mathcal{S}_{4}$ acts on the dual
$\mathbf{A}_{3}^{\vee}$ by means of the contragredient representation. Since
the reflection representation of $\mathcal{S}_{4}$ is self-dual over
$\mathbb{Q}$, these two representations are equivalent when tensored with
$\mathbb{Q}$. The direct sum $\mathbf{A}_{3}\oplus\mathbf{A}_{3}^{\vee}$ comes
with a natural quadratic form defined by $f(x,\xi):=\xi(x)$. This turns
$\mathbf{A}_{3}\oplus\mathbf{A}_{3}^{\vee}$ into a lattice isomorphic to
$\mathbf{U}^{\oplus 3}$ and thus produces a representation of the alternating
group $\mathcal{A}_{4}$ on $\operatorname{O}(\mathbf{U}^{\oplus 3})$.

After tensoring with $\mathbb{R}$, this representation becomes a direct sum of
two copies of the reflection representation, and so $\mathcal{S}_{4}$
preserves a positive-definite $3$-plane
$P\subset\mathbf{U}_{\mathbb{R}}^{\oplus 3}$. The subgroup
$\mathcal{A}_{4}\subset\mathcal{S}_{4}$ will act on $P$ in an orientation-
preserving manner. In order to extend this action to the $K3$ lattice
$\mathbf{U}^{\oplus 3}\oplus\mathbf{E}_{8}(-1)^{\oplus 2}$, note that there
exist two perpendicular copies of the $A_{3}$ root system in an $E_{8}$ root
system such that the associated embedding
$\mathbf{A}_{3}\oplus\mathbf{A}_{3}\hookrightarrow\mathbf{E}_{8}$ has the
property that its orthogonal complement is spanned by two mutually
perpendicular $(+4)$-vectors. So if we let $\mathcal{A}_{4}$ act on each
summand of $\mathbf{E}_{8}(-1)^{\oplus 2}$ through this embedding, then the
resulting action of $\mathcal{A}_{4}$ on the $K3$-lattice $\mathbf{U}^{\oplus
3}\oplus\mathbf{E}_{8}(-1)^{\oplus 2}\cong\mathbf{\Lambda}$ has the property
that $\mathbf{L}_{G}$ is spanned by $4$ mutually perpendicular $(-4)$-vectors.
Theorem 1.2 implies that $G$ is realized by a group of isometries of $M$
relative to some Ricci-flat metric, but cannot fix a complex structure. ∎

##  3 The fixed set of a diffeomorphism of prime order

The goal of this section is to prove some results on the structure of the
fixed-point set of a finite subgroup $\operatorname{Diff}(M)$ of prime order
$p$. In the first two subsections we introduce some basic tools we need in
order to do this.

Throughout this section, $N$ will be a connected, closed, oriented, smooth
$4$-manifold and $G\subset\operatorname{Diff}^{+}(N)$ will stand for a finite
cyclic group of prime order $p\geq 2$ of orientation-preserving
diffeomorphisms. Since $G$ is a finite group acting by diffeomorphisms, the
fixed set $N^{G}$ is an embedded, closed, smooth submanifold of $N$.

###  3.1 The Hirzebruch $G$-signature theorem

The first tool we will need is the Hirzebruch $G$-signature theorem, which we
now describe. The assumptions are as given in the previous paragraph.

Since the $G$-action on $N$ preserves orientation, each path component of
$N^{G}$ has dimension $0$ or $2$. At an isolated fixed point $z\in N^{G}$, the
$G$-action on the tangent space $T_{z}N$ preserves a complex structure that is
compatible with the given orientation, and is then given as a $2$-dimensional
complex $G$-representation by two nontrivial complex characters
$\chi_{1},\chi_{2}$, so equivalent to the complex representation

$g\in G:(z_{1},z_{2})\mapsto(\chi_{1}(g)z_{1},\chi_{2}(g)z_{2}).$ (3.1)

This is not unique: there is exactly one other complex structure with this
property and that is the conjugate of this one for which the associated
characters are of course $(\bar{\chi}_{1},\bar{\chi}_{2})$. Therefore the
signature defect ${\rm def}_{z}$ of the $G$-action at $z$ defined by

$\operatorname{def}_{z}:=\sum_{g\in
G\smallsetminus\\{1\\}}\frac{(1+\chi_{1}(g))(1+\chi_{2}(g))}{(1-\chi_{1}(g))(1-\chi_{2}(g))}.$
(3.2)

is intrinsic. Since $G$ is cyclic of prime order $p$, $\chi_{1}$ and
$\chi_{2}$ are generators of its group of complex characters and so
$\chi_{2}=\chi_{1}^{q}$ for some $q\in\mathbb{F}_{p}^{\times}$. We can then
also write this as

$\operatorname{def}_{z}=\sum_{\zeta\in\mu_{p}\smallsetminus\\{1\\}}\frac{(1+\zeta)(1+\zeta^{q})}{(1-\zeta)(1-\zeta^{q})}$

Note that each term in this sum is real, and is the norm squared of
$(1+\zeta)/(1-\zeta)$ when $q=-1$. For future reference we observe that the
Cauchy-Schwarz inequality implies that

$2\frac{(1+\zeta)(1+\zeta^{q})}{(1-\zeta)(1-\zeta^{q})}\leq\frac{(1+\zeta)(1+\zeta^{-1})}{(1-\zeta)(1-\zeta^{-1})}+\frac{(1+\zeta^{q})(1+\zeta^{-q})}{(1-\zeta^{q})(1-\zeta^{-q})},$

with equality holding only when
$(1+\zeta^{q})/(1-\zeta^{q})=(1+\zeta^{-1})/(1-\zeta^{-1})$, which simply
means that $q=-1$. It follows that $\operatorname{def}_{z}$ takes its maximal
value for $q=-1$ and for $q=-1$ only.

For each $2$-dimensional path component $C\subseteq N^{G}$, the signature
defect of the $G$-action at $Y$ is defined by (cf. [HZ], p.178, (15)):

$\operatorname{def}_{C}:=\frac{\displaystyle(p^{2}-1)}{\displaystyle 3}C\cdot
C,$ (3.2)

where $C\cdot C$ denotes the intersection product in $H_{2}(N;\mathbb{Z})$: it
is by definition zero unless $C$ is orientable, in which case we choose a
generator $[C]$ of the infinite cyclic group $H_{2}(C;\mathbb{Z})$ and then
take $[C]\cdot[C]$. So when $p=2$, we get $\operatorname{def}_{z}=0$ and
$\operatorname{def}_{C}=C\cdot C$.

Since $N/G$ is an oriented rational homology manifold, it has a well-defined
signature $\sigma(N/G)$, but for our purpose we can just as well define it as
the signature of the restriction of the intersection pairing to
$H_{2}(N;\mathbb{R})^{G}$.

######  Theorem 3.1 (Hirzebruch Signature Formula (see, e.g. (12) on page 177
of [HZ])).

Let $N$ be a closed, oriented $4$-manifold. Suppose that $G$ is a finite
subgroup of the group $\operatorname{Diff}^{+}(N)$ of orientation-preserving
diffeomorphisms of $N$ of prime order $p$. Then

$\textstyle p\cdot\sigma(N/G)=\sigma(N)+\sum_{z}{\rm def}_{z}+\sum_{C}{\rm
def}_{C},$ (3.3)

where $\sigma$ denotes the signature, the first sum ranges over the
$0$-dimensional path components $z$ of $N^{G}$, and the second sum ranges over
the $2$-dimensional path components $C$ of $N^{G}$.

###  3.2 $H_{2}(N;\mathbb{Z})$ as a $\mathbb{Z}[G]$-module

When $N$ is simply-connected, a result of Edmonds gives more precise
information about the $2$-dimensional components of $N^{G}$ in terms of the
$G$-action on $H_{2}(N;\mathbb{Z})$. In order to state this we will need a few
facts regarding $\mathbb{Z}[G]$-modules.

If $G$ is a finite cyclic group of prime order $p$, let
$\mathcal{I}_{G}\subset\mathbb{Z}[G]$ denote the augmentation ideal and let
$\check{\varepsilon}_{G}:=\sum_{g\in G}g\in\mathbb{Z}[G]$ be the co-
augmentation. Then
$\mathbb{Q}[G]=\mathbb{Q}\mathcal{I}_{G}\oplus\mathbb{Q}\check{\varepsilon}_{G}$
decomposes the algebra into a direct sum of the cyclotomic field of degree
$p-1$ (which we shall denote by $\mathbb{Q}(G)$) and $\mathbb{Q}$. Any
finitely-generated $\mathbb{Q}[G]$-module decomposes into a trivial module and
copies of the cyclotomic module $\mathbb{Q}(G)$.

There is an integral counterpart of this. First note that the image of
$\mathbb{Z}[G]$ in
$\mathbb{Q}(G)=\mathbb{Q}[G]/\mathbb{Q}\check{\varepsilon}_{G}$ is its ring of
integers, denoted here by $\mathbb{Z}(G)$. We shall refer to $\mathbb{Z}(G)$
as the _basic integral cyclotomic module_. As recalled by Edmonds in [Ed],
every finitely-generated module $H$ over the group-ring $\mathbb{Z}[G]$ which
is free abelian over $\mathbb{Z}$ decomposes into a trivial module and
_indecomposable_ summands that are either isomorphic to an ideal in
$\mathbb{Z}(G)$, called _integral cyclotomic_ or of $\mathbb{Z}$-rank $p$,
called _regular_. For the primes of interest here ($p=2,3,5,7$) the ideal
class group of $\mathbb{Q}(\mu_{p})$ is trivial, so that then every
indecomposable module is principal; that is, is isomorphic to $\mathbb{Z}$,
$\mathbb{Z}(G)$ or $\mathbb{Z}[G]$, respectively. Beware however that a
decomposition into isogeny types need not be unique.

######  Example 3.2.

A Coxeter transformation $c\in\operatorname{O}(\mathbf{E}_{8})$ is of order
$30$ and has as its eigenvalues the eight primitive $30$th roots of unity. So
for $p=2,3,5$, the transformation $c^{30/p}$ has prime order $p$ and does not
have $1$ as an eigenvalue. This makes $\mathbf{E}_{8}$ a purely cyclotomic
$\mathbb{Z}(\mu_{p})$-module; its rank as a $\mathbb{Z}(\mu_{p})$-module is
$8/(p-1)$, that is $8,4$ and $2$, respectively.

The obvious ring homomorphism
$\psi:\mathbb{Z}[G]\to\mathbb{Z}(G)\times\mathbb{Z}$ satisfies
$\psi(\check{\varepsilon}_{G})=(0,p)$ and of course $\psi(1)=(1,1)$. Thus
$\psi$ becomes a ring isomorphism if we invert $p$. In other words, a regular
$\mathbb{Z}[p^{-1}][G]$ module splits into a cyclotomic and a trivial module.
Put differently, the distinction between a regular module and a sum of a
cyclotomic module and a trivial module survives reduction modulo $p$. Indeed,
since a $p$th root of unity in an algebraic closure of $\mathbb{F}_{p}$ equals
$1$, tensoring with $\mathbb{F}_{p}$ renders the $G$-action unipotent and an
indecomposable regular resp. cyclotomic module is then given by a single
Jordan block of size $p$ resp. $p-1$. Here is simple, but useful observation.

######  Lemma 3.3.

Let $H$ be a free abelian group of finite rank and
$G\subset\operatorname{GL}(H)$ a subgroup of prime order $p$. Assume that as a
$\mathbb{Z}[G]$-module, $H$ has only regular and trivial summands and that the
ideal class group of $\mathbb{Q}(\mu_{p})$ is trivial. If $g\in G$ is a
generator, then the image of $g-1:H\to H$ is a direct summand of $H$ as a
$\mathbb{Z}$-module (and equal to the intersection $H^{\not=1}$ of $H$ with
the subspace of $H_{\mathbb{C}}$ where $g$ acts with eigenvalues $\not=1$).

If $H$ is endowed with a $G$-invariant unimodular form, then
$H^{\not=1}=(H^{G})^{\perp}$ and has discriminant lattice an
$\mathbb{F}_{p}$-vector space of dimension equal to the number of regular
summands in $H$.

###### Proof.

Our assumptions imply that $H$ is as a $\mathbb{Z}[G]$-module a direct sum of
copies of $\mathbb{Z}[G]$ and trivial $\mathbb{Z}[G]$-modules, and so it
suffices to check this for the case $H=\mathbb{Z}[G]$. But then the image of
$g-1$ is the augmentation ideal $\mathbf{I}_{G}\subset\mathbb{Z}[G]$ and is
indeed a direct summand.

Now assume that $H$ comes with $G$-invariant unimodular form and choose a
$\mathbb{Z}[G]$-linear isomorphism
$H=\mathbb{Z}[G]^{\nu}\oplus\mathbb{Z}^{\mu}$. The projector
$\frac{1}{p}\check{\varepsilon}_{G}$ defines orthogonal projection onto
$H^{G}_{\mathbb{Q}}$. It is clear that the image of this projector is
$\operatorname{\langle}\frac{1}{p}\check{\varepsilon}_{G}\operatorname{\rangle}^{\oplus\nu}\oplus\mathbb{Z}^{\mu}$.
Since $H$ is unimodular, the image of this projection can be identified with
$(H^{G})^{\vee}$. It follows that $(H^{G})^{\vee}/H^{G}$ is a
$\mathbb{F}_{p}$-vector space of dimension $\nu$. Since $H^{\not=1}$ is the
orthogonal complement of $H^{\perp}$ in $H$ its discriminant group is
canonically isomorphic to that of $H^{\perp}$. ∎

We can now state the following.

######  Proposition 3.4 (Edmonds, (Prop. (2.4) of [Ed])).

Let $N$ be a closed, oriented, simply-connected $4$-manifold and
$G\subset\operatorname{Diff}^{+}(N)$ a finite cyclic subgroup of prime order
$p$. Let $t,c$ and $r$ denote the number of trivial, cyclic and regular
summands of $H^{2}(N;\mathbb{Z})$ as a $\mathbb{Z}[G]$-module. Then

$b_{0}(N^{G};\mathbb{F}_{p})+b_{2}(N^{G};\mathbb{F}_{p})=t+2$

and

$b_{1}(N^{G};\mathbb{F}_{p})=c,$

where $b_{i}(N^{G};\mathbb{F}_{p})$ stands for the $i$th Betti number with
$\mathbb{F}_{p}$-coefficients,
$\dim_{\mathbb{F}_{p}}H^{i}(N^{G};\mathbb{F}_{p})$.

###  3.3 Cyclotomic summands and topology of the fixed set

In this subsection we concentrate on the case when $G$ is a cyclic subgroup of
order $p$ of $\operatorname{Diff}(M)$, with $p$ prime. We denote a generator
of $G$ by $f$.

Recall that a spin structure on a $4$-manifold $N$ is a lift of the structure
group $\operatorname{SO}(4)$ of the tangent bundle $TN$ to its universal
($2$-sheeted) cover $\widetilde{\operatorname{SO}}(4)$ (we will review this
notion in more detail in § 3.5).

######  Lemma 3.5 (Basic properties of $M^{G}$).

Let $G\subset\operatorname{Diff}(M)$ be a finite cyclic subgroup. Then:

  1. 1.

$G$ is orientation preserving.

  2. 2.

$M$ is a spin manifold and $G$ preserves a spin structure on $M$.

  3. 3.

$M^{G}$ is a finite union of points and closed, orientable $2$-manifolds.

###### Proof.

The first property was already noted in the introduction and appears here for
the sake of record only.

Since $M$ is a K3 manifold, $\pi_{1}(M)=0$ and $\omega_{2}(M)=0$. It follows
that $M$ has a unique spin structure, so that any diffeomorphism must preserve
this spin structure, proving (2).

By averaging any smooth Riemannian metric under the $G$ action, $G$ preserves
some Riemannian metric on $M$. It follows that $M^{G}$ is a (possibly
disconnected) closed manifold. Since $G$ is orientation-preserving, it acts on
the normal bundle of any component $C$ of $M^{G}$ as special orthogonal
transformation without eigenvalue $1$. This implies that such a component has
even dimension (it is a finite union of points and closed $2$-manifolds). This
argument also shows that if $C$ is a surface component and $G$ has odd order,
then this normal bundle must be orientable (so that $C$ is orientable). If
$p=2$ then Proposition 3.2 of [Ed] implies, since $G$ is orientation-
preserving and spin-preserving, that $M^{G}$ is orientable.(333In this and in
other results quoted from [Ed], the hypothesis is actually that the $G$-action
is locally linear. This is satisfied for any diffeomorphism.) This proves (3).
∎

Denote by $\nu$ the multiplicity of the cyclotomic character of the $G$-action
on $H_{2}(M;\mathbb{Q})$. Note that the cyclotomic character has degree $p-1$,
which implies that

$\nu(p-1)\leq 22-3=19.$

######  Proposition 3.6 (Topology of $M^{G}$).

Let $G$ be cyclic subgroup of prime order $p$ of the diffeomorphism group of
the K3 manifold $M$ which pointwise fixes a positive $3$-plane in
$H_{2}(M;\mathbb{R})$. Denote by $\nu$ the multiplicity of the rational
cyclotomic character of the $G$-action on $H_{2}(M;\mathbb{Q})$. Then the
following hold:

  1. 1.

The Euler characteristic of $M^{G}$ is

$\chi(M^{G})=24-\nu p.$ (3.4)

  2. 2.

The signature of $M/G$ is

$\sigma(M/G)=\sigma(H_{2}(M;\mathbb{R})^{G})=\nu(p-1)-16.$

  3. 3.

The signature defect of the $G$-action on $M$ is

$\textstyle\sum_{z}{\rm def}_{z}+\sum_{C}{\rm def}_{C}=(p-1)(\nu p-16).$ (3.5)

###### Proof.

Since $H_{2}(M;\mathbb{R})\cong\mathbb{R}^{22}$ decomposes into $\nu$
cyclotomic summands (each of which has dimension $p-1$) and $22-(p-1)\nu$
trivial summands, and since the character of each cyclotomic summand equals
$-1$, it follows that the Lefschetz number of a generator $f$ of $G$ is

$L(f)=1+1\cdot(22-(p-1)\nu)+(-1)\cdot\nu+1=24-\nu p.$

But $\chi(M^{G})=\chi(M^{f})=L(f)$, the first equality following since $f$
generates $G$; the second following from the Lefschetz Fixed-Point Theorem.
This proves (1).

As noted in § 3.2, $H_{2}(M;\mathbb{Q})$ decomposes as a direct sum of
$1$-dimensional trivial and $(p-1)$-dimensional cyclotomic
$G$-representations. By definition there are $\nu$ and so $22-\nu(p-1)$ of the
former. Since we assumed that $H_{2}(M;\mathbb{R})^{G}$ contains positive-
definite $3$-plane, it follows that the intersection form on
$H_{2}(M;\mathbb{R})$ restricts to a bilinear form on
$H_{2}(M;\mathbb{R})^{G}$ of type $(3,19-\nu(p-1))$. Thus
$\sigma(H_{2}(M;\mathbb{R})^{G})=\nu(p-1)-16$, proving (2).

Given this, the $G$-signature Formula (Theorem 3.1) implies

$\textstyle\sum_{z}{\rm def}_{z}+\sum_{C}{\rm
def}_{C}=p(-16+\nu(p-1))+16=(p-1)(\nu p-16),$

proving (3). ∎

###  3.4 Smooth involutions have finite fixed set

In this subsection we focus on the case $p=2$. _Throughout this section we
also assume that $G$ fixes pointwise a positive-definite $3$-plane in
$\mathbf{\Lambda}$._ This implies that $\mathbf{L}_{G}$ is the orthogonal
complement of $H^{2}(M)^{G}$ in $H^{2}(M)$ and that $\mathbf{L}_{G}$ is
negative definite.

The first part of the following proposition is due to Bryan ([Bry], Section
4.2) and was obtained by him in a somewhat different manner. It has in common
with our proof that it is based on Seiberg-Witten theory.

######  Proposition 3.7.

Let $G\subset\operatorname{Diff}(M)$ be a subgroup of order $2$. Then the
fixed set $M^{G}$ consists of $8$ points and the $\mathbb{Z}[G]$-module
$H_{2}(M)$ has $8$ regular summands, $6$ trivial summands and no cyclotomic
summands.

###### Proof.

Let $P\subset H^{2}(M;\mathbb{R})$ a positive-definite $3$-plane pointwise
fixed by $G$. By Lemma 3.5, $M^{G}$ is a finite union of points and closed
surfaces. That lemma also tells us that the $G$-action preserves a spin
structure on $M$. A result of Atiyah-Bott ([AB], Proposition 8.46) states that
then all components of $M^{G}$ have the same dimension: $0$ or $2$. We first
exclude the last case. Suppose otherwise and assume that $M^{G}$ is a finite
union $C_{1}\cup\cdots\cup C_{m}$ of closed, connected surfaces, $m\geq 1$.

Since all components of $M^{G}$ have the same dimension, Corollary 2.6 of
Edmonds [Ed] implies that the classes of any $m-1$ elements of
$\\{C_{1},\ldots,C_{m}\\}$ in $H_{2}(M;\mathbb{F}_{2})$ span an
$(m-1)$-dimensional subspace. In particular each $C_{i}$ defines a nonzero
class in $H_{2}(M;\mathbb{F}_{2})$. Since $C_{i}$ is orientable by Lemma 3.5,
it will then even support a nonzero integral class.

Formulas (1) and (3) of Proposition 3.6 give $\sum_{i}\chi(C_{i})=24-2\nu$ and
$\sum_{i}C_{i}\cdot C_{i}=2\nu-16$, and so they combine to give

$\sum_{i}\big{(}\chi(C_{i})+C_{i}\cdot C_{i}\big{)}=8$ (3.6)

We claim that each term on the left-hand side of (3.6) is $\leq 0$; this
evidently gives a contradiction. If $C_{i}\cdot C_{i}<0$, then the fact that
the intersection form on $M$ is even implies that $C_{i}\cdot C_{i}\leq-2$,
and hence $\chi(C_{i})+C_{i}\cdot C_{i}\leq 0$. On the other hand, when
$C_{i}\cdot C_{i}\geq 0$, then as we have seen, $C_{i}$ supports a nonzero
integral homology class, and then the adjunction inequality of Kronheimer-
Mrowka (Theorem 1.1 of [KM]) asserts that it is then still true that
$\chi(C_{i})+C_{i}\cdot C_{i}\leq 0$, as desired.

Now that we know that $M^{G}$ is finite, we invoke the signature defect
formula 3.6-(3). It tells us that $0=2\nu-16$, so that $\nu=8$. Substituting
this in the Lefschetz formula 3.6-(1) then gives that $M^{G}$ consists of $8$
points. Proposition 3.4 shows that the $\mathbb{Z}[G]$-module invariants of
$H_{2}(M)$ cannot have any cyclotomic summands, and so the number of regular
and trivial summands will be as asserted. ∎

Proposition 3.7 is already enough to prove Corollary 1.10.

###### Proof of Theorem 1.10.

First note that a Dehn twist $T_{S}\in\operatorname{Diff}(M)$ induces an
element of $O(H_{M})$ that is reflection in a $(-2)$-vector. Suppose that
$g:M\to M$ is a diffeomorphism of finite order such that $g_{*}=(T_{S})_{*}$.
If $\operatorname{\langle}g\operatorname{\rangle}$ denotes the group generated
by $g$, then the existence of the natural surjection
$\operatorname{\langle}g\operatorname{\rangle}\to\operatorname{\langle}g_{*}\operatorname{\rangle}\cong\mathbb{Z}/2\mathbb{Z}$
implies that the order of $g$ is even, say $2m$, $m\geq 1$. Then $g^{m}$ is a
smooth involution. Since $g_{*}=(T_{S})_{*}$ has order $2$ then either
$g^{m}_{*}$ equals $(T_{S})_{*}$, so that $g^{m}\in W_{G}$, or that
$(g^{m})_{*}$ is the identity. Since $g^{m}$ is a smooth involution which
evidently does not satisfy the conclusion of Proposition 3.7, we get a
contradiction. ∎

Corollary 4.12 gives a more precise description of how an involution acts on
$M$.

###  3.5 Resolution of a spin orbifold of dimension 4

Let $G\subset\operatorname{Diff}(M)$ be any finite group. When $M^{G}$ has an
isolated fixed point $p$ then the quotient $M/G$ is not a manifold, since a
neighborhood of the image of $p$ in $M/G$ is the cone on a lens space. The
goal of this section is to explain how one can sometimes resolve $M/G$ to
obtain a new smooth manifold $M^{\prime}$ equipped with a smooth $G$-action.
We then deduce properties of the original action from properties of
$M^{\prime}$.

We begin with a general discussion of how to construct spin structures on
certain resolutions. We will apply this to the resolved quotient of a K3
surface by a finite group of prime order.

Spin structures. Recall that a spin structure on an oriented $4$-manifold $W$
amounts to a lift of the structural group of its tangent bundle to the
universal cover of $\textrm{SL}_{4}(\mathbb{R})$ (which has degree $2$). In
other words, if $\textrm{SL}_{W}\to W$ denotes the bundle of oriented bases of
the tangent spaces of $W$ (this is a principal
$\textrm{SL}_{4}(\mathbb{R})$-bundle), then a spin structure is given by a
double cover $\widetilde{\textrm{SL}}_{W}\to\textrm{SL}_{W}$, called a _spinor
bundle_ , which defines a connected (and hence universal) cover of each
$\textrm{SL}_{4}(\mathbb{R})$-orbit. A necessary and sufficient condition for
its existence is the vanishing of the second Stiefel-Whitney class $w_{2}(W)$
of the tangent bundle. It is unique up to isomorphism if and only if the first
Stiefel-Whitney class also vanishes. We note here that if $W$ is closed then
the existence of a spin structure on $W$ implies that its intersection form is
even: this follows from a special case of Wu’s formula, which states that
$x\cup x=w_{2}(W)\cup x$ for all $x\in H^{2}(W;\mathbb{F}_{2})$ (see for
instance [MS]).

Let $G$ be a finite subgroup of $\operatorname{Diff}(W)$ that preserves the
spin structure in the sense that every element of $G$ lifts to the spinor
bundle $\widetilde{\textrm{SL}}_{W}\to W$. We say that $G$ is _even_ if $G$
admits a lift _as a group_ to $\widetilde{\textrm{SL}}_{W}$ (so that there is
no need to extend $G$ with the covering transformation of
$\widetilde{\textrm{SL}}_{W}\to\textrm{SL}_{W}$).

Suppose $W$ is a complex manifold endowed with a nowhere zero holomorphic
$2$-form $\omega$. Then there is a reduction of the structural group of the
tangent bundle to $\textrm{SL}_{2}(\mathbb{C})$ by letting
$\textrm{SL}(\mathbb{C})_{W}$ be the bundle of complex bases on which $\omega$
takes the value $1$. Since $\textrm{SL}_{2}(\mathbb{C})$ is simply connected,
the inclusion $\textrm{SL}_{2}(\mathbb{C})\subset\textrm{SL}_{4}(\mathbb{R})$
lifts canonically to an embedding of $\textrm{SL}_{2}(\mathbb{C})$ in the
universal cover of $\textrm{SL}_{4}(\mathbb{R})$, and hence this gives rise to
a spin structure on $W$. So a finite group of diffeomorphisms of $W$ that
preserves both the complex structure and the form $\omega$ comes with a lift
to the associated spinor bundle: it is even.

The resolved orbit space. In what follows we consider the following special
case: $G$ is a nontrivial finite subgroup of $\textrm{SL}_{2}(\mathbb{C})$ and
$W$ is an open contractible $G$-invariant neighborhood of the origin in
$\mathbb{C}^{2}$, for example the unit ball with respect to a $G$-invariant
inner product. Note that this contains the important case when $G$ is of order
two, its nontrivial element being the antipodal involution. We observe that
$G$ then also leaves $(dz_{1}\wedge dz_{2})|_{W}$ invariant. The group $G$
acts freely on $W\smallsetminus\\{0\\}$ and so the $G$-orbit space $W_{G}$ of
$W$ has, as a (normal) singular point, the image of the origin, which we shall
denote by $0_{G}$; topologically it is the open cone over the quotient of a
$3$-sphere by a free $G$-action.

This singularity has a well known resolution $\pi:W^{\prime}\to W_{G}$. Here
$W^{\prime}$ is a complex manifold of dimension $2$, the map $\pi$ is
holomorphic and is an isomorphism over $W_{G}\smallsetminus 0_{G}$ and
$D:=\pi^{-1}(0_{G})$ is a normal crossing curve whose irreducible components
are all copies of $\mathbb{P}^{1}$ having self-intersection $-2$ (we call such
an irreducible component a _$(-2)$ -curve_). They meet each other
transversally in such a manner that the dual intersection graph is the Dynkin
diagram of type $A$, $D$ or $E$.

The case that interests us most is when $G$ is cyclic of order $n$, in which
case $D$ is a chain of $(-2)$-curves, $n-1$ in number, so that the dual
intersection diagram is of type $A_{n-1}$. Moreover, the holomorphic form
$\omega:=(dz_{1}\wedge dz_{2})|_{W}$, which because of its $G$-invariance
descends to a nowhere zero holomorphic $2$-form on $W_{G}\smallsetminus
0_{G}$, extends across $W^{\prime}$ as a _nowhere-zero holomorphic $2$-form
$\omega^{\prime}$_. This proves that the
$\textrm{SL}_{2}(\mathbb{C})$-reduction of the tangent bundle of the $G$-orbit
space of $W\smallsetminus\\{0\\}$ (which is just $W^{\prime}\smallsetminus D$)
extends to $W^{\prime}$. In particular, $W^{\prime}$ comes with a spin
structure.

Conversely, suppose that we are given a pair $(W^{\prime},D)$, where $D$ is a
normal crossing divisor that is a deformation retract of the complex surface
$W^{\prime}$ and is of the type above: a finite union of $(-2)$-curves meeting
transversally so that the dual intersection graph is a Dynkin diagram of type
$A$, $D$ or $E$. Then $G=\pi_{1}(W^{\prime}\smallsetminus D)$ is finite. The
universal cover of $W^{\prime}\smallsetminus D$ is a complex surface that
extends over $W^{\prime}$ as a complex normal variety that has singularities
over the singular part $D_{\mathit{sg}}$ (the set of normal crossing points)
of $D$. These can be resolved in a minimal manner to produce a complex
manifold $\tilde{W}$ and a holomorphic map $\pi^{\prime}:\tilde{W}\to
W^{\prime}$ such that $\pi^{\prime}$ is a double cover over
$W^{\prime}\smallsetminus D_{\mathit{sg}}$ and the preimage $\tilde{D}$ of $D$
is a normal crossing divisor.

The normal crossing divisor $\tilde{D}$ is simply-connected: its irreducible
components are copies of $\mathbb{P}^{1}$ and the dual intersection graph is a
tree. In fact, $\tilde{D}$ contains at least one exceptional curve of the
first kind (a _$(-1)$ -curve_), and if we contract such curves and keep on
doing that until none are left to contract, then we have in fact contracted
all of $\tilde{D}$. We thus obtain a proper holomorphic map $\pi:\tilde{W}\to
W$ to a complex surface which maps $\tilde{D}$ to a singleton $\\{o\\}$ and is
an isomorphism over $W\smallsetminus\\{o\\}$. The naturality of this
construction guarantees that the $G$-action extends to $\tilde{W}$ and
descends to $W$ with $o$ as its unique fixed point. A nowhere zero holomorphic
$2$-form $\omega^{\prime}$ on $W^{\prime}$ determines one on $W$ which is
$G$-invariant and this identifies $G$ with a finite subgroup of
$\textrm{SL}(T_{o}W)$.

Note that if $n=2$, then $D$ is a single $(-2)$-curve and the construction of
$\tilde{W}$ requires no blowing up: it is just a double cover of $W^{\prime}$
ramified over $D$ and the preimage $\tilde{D}$ of $D$ is a $(-1)$-curve that
can be blown down to produce the surface $W$ on which $G$ acts as an antipodal
involution. In a global setting this construction yields the following.

######  Corollary 3.8 (Resolved orbit space).

Let $W$ be an oriented $4$-manifold. Let $G$ be a finite subgroup of
$\operatorname{Diff}^{+}(W)$ such for every $p\in W$ the stabilizer $G_{p}$
acts in some neighborhood of $p$ by an $\textrm{SL}_{2}(\mathbb{C})$-model, in
the sense that there is a $\mathbb{C}^{2}$-valued oriented chart in terms of
which the stabilizer $G_{p}$ acts as a subgroup of
$\textrm{SL}_{2}(\mathbb{C})$. Then the set $F\subset W$ of $p\in W$ for which
$G_{p}\not=\\{1\\}$ is discrete. Further, there is a diagram in the category
of $G$-manifolds endowed with a $G$-invariant union of submanifolds that meet
transversally:

$(W,F)\xleftarrow{\pi_{W}}(\tilde{W},\tilde{D})\xrightarrow{\pi_{W^{\prime}}}(W^{\prime},D)$

and it is obtained locally as above:
$\pi_{W}^{-1}F=\tilde{D}=\pi_{W^{\prime}}^{-1}D$ with $G$ acting trivially on
$W^{\prime}$ so that

$W\smallsetminus F\xleftarrow{\cong}\tilde{W}\smallsetminus\tilde{D}\to
W^{\prime}\smallsetminus D$

is the formation of the $G$-orbit space of $W\smallsetminus F$. Both
$\tilde{D}$ and $D$ admit neighborhoods and complex structures in terms of
which they become normal crossing divisors of the type described above. If $W$
comes with a $G$-invariant spin structure then so does $W^{\prime}$, and vice
versa.

###### Proof.

At every $p\in W$ with $G_{p}\not=\\{1\\}$, we choose a
$\mathbb{C}^{2}$-valued chart $\kappa:U\hookrightarrow\mathbb{C}^{2}$ that is
orientation preserving and $G_{p}$-equivariant relative to an embedding
$G_{p}\subset\textrm{SL}_{2}(\mathbb{C})$. The given spin structure on this
chart is trivial and the preceding discussion shows that the resulting spin
structure on the $G_{p}$-orbit space of $U\smallsetminus\\{p\\}$ extends
(uniquely) across the resolution of the $G_{p}$-orbit space of $U$. ∎

##  4 Prime order subgroups of $\operatorname{Diff}(M)$

Nikulin determined [Nik1] which abelian groups $G$ can appear as automorphism
groups of some complex K3 surface $X$ that act trivially on $H^{2,0}(X)$. In
our setting these are precisely the abelian subgroups of
$\operatorname{Diff}(M)$ that fix a positive $3$-plane in
$\mathbf{\Lambda}_{\mathbb{R}}$ and some complex structure on $M$. If we
assume that $G$ is cyclic of prime order $p$, then the primes $p$ that occur
are $2,3,5,7$. A more detailed study for the case $p=2$ was given by Morrison
[Mor] and Van Geemen-Sarti [vGS] and for the odd primes by Garbagnati-Sarti
[GS]. There is in fact a very helpful picture in terms of an elliptic
fibration that enables us to get a good understanding of the topology of such
an action. We will say more about this in Section 4.7.

In this section $G\subset\operatorname{Diff}(M)$ has prime order $p\geq 2$.
_Throughout this section we also assume that $G$ fixes pointwise a positive-
definite $3$-plane in $\mathbf{\Lambda}_{\mathbb{R}}=H_{2}(M;\mathbb{R})$._
This implies that $\mathbf{L}_{G}$ is the orthogonal complement of
$\mathbf{\Lambda}^{G}$ in $\mathbf{\Lambda}$ and that $\mathbf{L}_{G}$ is
negative definite.

###  4.1 A first reduction for the case $p$ odd

Let $R_{G}$ be the set of $(-2)$-vectors in $\mathbf{L}_{G}$. Since
$\mathbf{L}_{G}$ is negative definite, $R_{G}$ is finite. It is in fact a root
system whose roots are all of the same length. This implies that its
irreducible components are of type $A$, $D$ or $E$. We show here that there
are only two possibilities which are in a sense opposite extremes.

######  Theorem 4.1 (Dichotomy).

Let $G\subset\operatorname{Diff}(M)$ be a group of prime order $p\geq 3$, and
suppose that $G$ fixes pointwise a positive-definite $3$-plane in $H_{2}(M)$.
Assume that $H_{2}(M)$ has as a $\mathbb{Z}[G]$-module no cyclotomic summands.
If $\nu$ is the number of regular summands then $\nu(p+1)\leq 24$ with
equality holding if and only if $(p,\nu)\in\\{(3,6),(5,4),(7,3)\\}$. In this
case $M^{G}$ consists of $\nu$ points with the $G$ action on the tangent space
at each point factoring through a representation to
$\textrm{SL}_{2}(\mathbb{C})$.

Furthermore, precisely one of the following holds:

(Nikulin type)

$R_{G}=\emptyset$ and the above inequality is an equality. Moreover there is a
possibly different action of $G$ on $M$ that acts on $H_{2}(M)$ as the given
one and which preserves a complex structure and Ricci-flat Kähler metric $M$
and a nonzero holomorphic $2$-form on the resulting complex surface.

(Coxeter type)

$R_{G}$ is a subsystem of type $A_{p-1}^{\nu}$, spans $\mathbf{L}_{G}$ and $G$
acts in each $A_{p-1}$-summand as a Coxeter transformation of that summand.

######  Remarks 4.2.

  1. 1.

In the ‘Nikulin type’ case of Theorem 4.1 we do not know whether the given
$G$-action preserves a Ricci-flat Kähler structure, but as will become clear
below, we can come close. We also suspect that the Coxeter case never occurs.

  2. 2.

It follows from Corollary 2.2 of Chen-Kwasik [CK] that if $G$ preserves a
symplectic form then $M^{G}$ is finite and $G$ acts in the tangent space of
each fixed point through $\textrm{SL}_{2}(\mathbb{C})$. It then follows from
Proposition 3.4 that $H_{2}(M)$ has as a $\mathbb{Z}[G]$-module no cyclotomic
summands. So in this case Theorem 4.1 implies that $\nu(p+1)=24$, so that the
dichotomy there described holds.

We now turn to the proof of Theorem 4.1.

###### Proof of Theorem 4.1.

The proof is in three steps.

_Step 1: The action of $G$ on $R_{G}$ is via an element of the Weyl group
$W(R_{G})$._

Let $v\in R_{G}$ and let $g\in G\smallsetminus\\{1\\}$. Then $gv\not=\pm v$
and so $v$ and $gv$ are linearly independent. They therefore span a negative
sublattice of rank $2$. Since $v\cdot v=gv\cdot gv=-2$, it follows that
$v\cdot gv\in\\{-1,0,1\\}$. Since $\mathbf{L}_{G}$ has no nonzero $G$-fixed
vectors, we must have $\sum_{g\in G}gv=0$. We cannot have $v\cdot gv=0$ for
all $g\in G\smallsetminus\\{1\\}$, for then the $G$-orbit of $v$ consists of
pairwise orthogonal vectors and this contradicts $\sum_{g\in G}gv=0$. So for
some $g\in G$, we will have $v\cdot gv=1$ and it then follows that $Gv$
generates an irreducible root system of rank $\leq p-1$. (This must be of rank
exactly $p-1$ with $G$ acting in a standard manner as there are no nontrivial
$\mathbb{Q}[G]$-modules of lower rank.)

Let $R_{o}$ be an irreducible component of $R_{G}$. From what we just proved
it follows that $G$ preserves $R_{o}$. We claim that $G$ acts on $R_{o}$ as an
element of the Weyl group $W(R_{o})$. If this were not so then it would act
nontrivially on the Dynkin diagram of $R_{o}$. The Dynkin diagrams of type
$A$, $D$ or $E$ with a nontrivial automorphism are $A_{r}$ ($r\geq 2$),
$D_{r}$ ($r\geq 4$) and $E_{6}$. For all but $D_{4}$ this is an automorphism
group is of order $2$ (and so this cannot happen since we assumed $p$ odd) and
for $D_{4}$ and the group in question is isomorphic to $\mathcal{S}_{3}$. This
last case cannot happen either, because then $p=3$ and any order $3$
automorphism of a $D_{4}$ root system which is not in the Weyl group permutes
three mutually perpendicular roots and hence will fix a vector.

_Step 2: $\nu(p+1)\leq 24$ and equality holds if and only if
$(p,\nu)\in\\{(3,6),(5,4),(7,3)\\}$ and $M^{G}$ consists of $\nu$ points with
$G$ acting in the tangent space of each such point through
$\textrm{SL}_{2}(\mathbb{C})$._

$M^{G}$ is a union of $m\geq 0$ isolated points $z_{1},\dots,z_{m}$ and $r\geq
0$ connected, embedded subsurfaces $C_{1},\dots,C_{r}$. Formulas (1) and (3)
of Proposition 3.6 give

$\displaystyle m+\sum_{j=1}^{r}\chi(C_{j})=24-\nu p,$
$\displaystyle\frac{1}{p-1}\sum_{i=1}^{m}\operatorname{def}_{z_{i}}+\frac{p+1}{3}\sum_{j=1}^{r}C_{j}\cdot
C_{j}=\nu p-16,$

and so they combine to give

$m+\frac{1}{p-1}\sum_{i=1}^{m}\operatorname{def}_{z_{i}}+\sum_{j=1}^{r}\big{(}\chi(C_{j})+\frac{p+1}{3}C_{j}\cdot
C_{j}\big{)}=8.$ (4.1)

Since by assumption $H_{2}(M)$ has as a $\mathbb{Z}[G]$-module no cyclotomic
summands, Proposition 3.4 implies that each $C_{j}$ has genus zero. In other
words, $\chi(C_{j})=2$ for each $j$. As we have seen, each $C_{j}$ supports a
nonzero integral homology class, so that by the adjunction inequality of
Kronheimer-Mrowka (Theorem 1.1 of [KM]), $C_{j}\cdot C_{j}\leq-2$. Hence each
term of the second sum on the left-hand side of (4.1) is $\leq 0$. It follows
that

$m+\frac{1}{p-1}\sum_{i=1}^{m}\operatorname{def}_{z_{i}}\geq 8.$

As we already observed in § 3.1, the maximal value of
$\operatorname{def}_{z_{i}}$ is assumed in the case where the action
$G$-action on $T_{z_{i}}M$ factors through $\textrm{SL}_{2}(\mathbb{C})$; this
value is as Chen-Kwasik note (Lemma 2.4 of [CK]) equal to
$\tfrac{1}{3}(p-1)(p-2)$. Thus we get the inequality

$m+\frac{1}{3}(p-2)m\geq 8,\text{ or equivalently, }(p+1)m\geq 24.$

On the other hand, $m+\sum_{j=1}^{r}\chi(C_{j})=24-\nu p$ and so $m+\nu p\leq
24\leq(p+1)m$, which implies that $\nu\leq m$ and hence that $\nu(p+1)\leq
24$.

Suppose we have equality: $\nu(p+1)=24$. Then it follows that $\nu=m$ and all
the equalities above are equalities: $\sum_{j=1}^{r}(2+\frac{p+1}{3}C_{j}\cdot
C_{j})=0$ and since $C_{j}\cdot C_{j}\leq-2$, this means that $r=0$ (and so
$M^{G}$ finite) and $(p,\nu)\in\\{(3,6),(5,4),(7,3)\\}$ and
$\operatorname{def}_{z_{i}}=\tfrac{1}{3}(p-1)(p-2)$ for all $i=1,\dots,m$,
which means that $G$ acts in $T_{z_{i}}M$ through
$\textrm{SL}_{2}(\mathbb{C})$.

_Step 3: Proof of the dichotomy._

Choose a generator $g\in G$ and let $w\in W(R_{G})$ be such that $w$ acts on
$R_{G}$ as $g_{*}$. Then $g_{*}$ and $w$ commute so that either $g_{*}w^{-1}$
has still order $p$ in $\operatorname{O}^{+}(\mathbf{\Lambda})$ or $g_{*}=w$.
In the first case, $g_{*}w^{-1}$ generates a subgroup
$G_{0}\subset\operatorname{O}^{+}(\mathbf{\Lambda})$ of order $p$ with the
property that $L_{G_{0}}$ contains no $(-2)$-vectors. According to Nikulin
[Nik1], this can only happen when $L_{G_{0}}$ has rank $(p-1)\nu_{0}$ with
$(p,\nu_{0})\in\\{(3,6),(5,4),(7,3)\\}$. In all these cases $(p+1)\nu_{0}=24$.
It is also clear that $\nu_{0}\leq\nu$. But since $(p+1)\nu\leq 24$ (by step
1), it then follows that $\nu_{0}=\nu$. So then
$\mathbf{L}_{G}=\mathbf{L}_{G_{0}}$, meaning that $R_{G}=\emptyset$ and the
action of $G$ being of Nikulin type.

In the second case, we are given that $g$ is contained in a finite reflection
group $W$. We then write $g_{*}=r_{v_{1}}\circ\cdots\circ r_{v_{m}}$ for
certain $(-2)$-vectors $v_{1},\dots,v_{m}$ in $R_{G}$. According to Lemma 3 of
[Car], we then may take the $v_{i}$ to be linearly independent, so that
$m=\nu(p-1)$.

The root system generated by the $v_{1},\dots,v_{\nu(p-1)}$ has only summands
of type $A$, $D$ or $E$ and its Weyl group contains $g_{*}$ as an element of
prime order $p$. R.W. Carter’s Table 3 in [Car] tells us that such an element
is given by a Coxeter transformation of a root subsystem $R_{0}$, all of whose
components are of type $A_{p-1}$. We are given that $\mathbf{L}_{G}$ is
contained in a $\mathbb{Z}[G]$-submodule of $\mathbf{\Lambda}$ that is
isomorphic to $\mathbb{Z}[G]^{\nu}$. This means that there exist
$v_{1},\dots,v_{\mu}$ in $\mathbf{\Lambda}$ such that $L_{G}$ has basis

$\\{g^{j}(g-1)v_{i}\,:\,i=1,\dots,\nu;j=1,\dots,p-1\\}.$

Since $R_{0}$ generates a sublattice of $\mathbf{L}_{G}$ of finite index, this
can only happen when each $(g-1)v_{i}$ is a $(-2)$-vector. But then it is
clear that the above basis must be a root basis of $R_{0}$. In particular
$R_{0}=R_{G}$ spans $\mathbf{L}_{G}$. ∎

The content in the rest of Section 4. Of the two possibilities in the
dichotomy of Theorem 4.1, only the Coxeter case is explicit. The other, more
interesting case, is when $\mathbf{L}_{G}$ contains no $(-2)$-vector. Since
its existence was established by Nikulin, this lattice has been the subject of
further study (see below for references). Yet such studies have mostly been on
a case-by-case basis, sometimes aided with computer calculations.

Our take here is somewhat different: it is entirely uniform (it is not based
on a case-by-case treatment and includes the case $p=2$) and leads to a novel
characterization of lattices of Nikulin type (see Proposition 4.9ff). This
will be the key to our proof that the finite prime order subgroups of
$\operatorname{Diff}(M)$ which leave invariant a complex structure make up a
single conjugacy class (Theorem 1.6). At the same time we analyze the
homological aspects of the resolved orbit space construction. Indeed we arrive
at the candidate for $\mathbf{L}_{G}$ via this construction which involves
another lattice that we shall refer to as the _Nikulin lattice_ (for $p=2$
this was exhibited by Morrison and was thus called by Van Geemen-Sarti). This
lattice will also help us to identify the characters of $G$ associated to its
representation on the tangent space of a fixed point. We will thus see how
close we come to eliminating the Coxeter type in Theorem 4.1. We shall see for
$p=2$ that we will always have a Nikulin-type situation and indeed, we will
find that then the answer is: quite close.

###  4.2 The resolved orbifold $M_{G}$

We assume in this subsection that if $p$ is odd then $G$ acts at each fixed
point according to an $\textrm{SL}_{2}(\mathbb{C})$-model (when $p=2$ then
$M^{G}$ is finite by Proposition 3.7, so this is then always the case). We
denote by $\nu$ the number of rational cyclotomic summands in
$\mathbf{\Lambda}_{\mathbb{Q}}$. This is then also the number of fixed points
of $G$ in $M$ and we have $(p+1)\nu=24$ (either by Proposition 3.7 or Theorem
4.1). We begin by applying the construction of Corollary 3.8.

######  Proposition 4.3.

Let $p\geq 2$. When $p>2$ assume that the group $G$ acts at each fixed point
according to an $\textrm{SL}_{2}(\mathbb{C})$-model. Then
$\mathbf{L}_{G}^{\vee}/\mathbf{L}_{G}$ is an $\mathbb{F}_{p}$-vector space of
dimension $\nu$. Further, if we form the resolved orbit space as in Corollary
3.8:

$(M,M^{G})\xleftarrow{\pi_{M}}(\tilde{M},\tilde{D})\xrightarrow{\pi_{M^{\prime}}}(M^{\prime},D),$

then $M^{\prime}$ is homeomorphic to a $K3$ manifold and the rank of
$H_{2}(M^{\prime}\smallsetminus D)$ is $2\nu-2$.

###### Proof.

It is clear that $M^{\prime}$ is a oriented closed $4$-manifold. It is simply-
connected since the inclusion $M^{\prime}\smallsetminus D\subset M^{\prime}$
induces a surjection on fundamental groups; $\pi_{1}(M^{\prime}\smallsetminus
D$) is cyclic of order $p$ and a generator will die in $M^{\prime}$ near a
point in $D$ of total ramification.

Since the irreducible components of $D$ have an intersection matrix that is
negative definite, the map $H_{2}(D;\mathbb{Q})\to
H_{2}(M^{\prime};\mathbb{Q})$ is injective and has image a negative-definite
subspace of dimension $\nu(p-1)$ (there are $\nu$ pairwise disjoint copies of
chains of length $p-1$.) Its orthogonal complement is the image of
$H_{2}(M^{\prime}\smallsetminus D^{\prime};\mathbb{Q})$. The transfer map
identifies $H_{2}(M^{\prime}\smallsetminus D;\mathbb{Q})$ with
$H_{2}(M\smallsetminus M^{G};\mathbb{Q})^{G}$ and multiplies the intersection
pairing with $p$. The latter has signature $(3,19-(p-1)\nu)$. Hence
$H_{2}(M^{\prime})$ has signature $(3,19-(p-1)\nu+(p-1)\nu)=(3,19)$. Since $M$
has a unique spin structure, $G$ must leave that structure invariant and so
$M^{\prime}$ inherits a spin structure. Wu’s formula then implies that the
intersection form on $M^{\prime}$ is even. Freedman’s classification of
closed, simply-connected $4$-manifolds (Theorem 1.5 of [Fr]) then implies that
$M^{\prime}$ must be homeomorphic to a $K3$ manifold.

The rank of $H_{2}(M^{\prime}\smallsetminus D)$ is $22$ minus the rank of
$H_{2}(D)$. Since the latter is $\nu(p-1)$, we get $22-\nu
p+\nu=(-2+\nu)+\nu=2\nu-2$. ∎

We continue with the situation of Proposition 4.3. Consider part of the exact
sequence of the Gysin sequence for the pair $(M^{\prime},D)$:

$H^{1}(M^{\prime}\smallsetminus D)\to H_{2}(D)\to H^{2}(M^{\prime})\to
H^{2}(M^{\prime}\smallsetminus D)\to H_{1}(D).$

The first and the last term are zero: since $\pi_{1}(M^{\prime}\smallsetminus
D)=G$ it follows that $H^{1}(M^{\prime}\smallsetminus
D)=\operatorname{Hom}(G,\mathbb{Z})=0$ and the simple-connectivity of $D$
implies that $H_{1}(D)=0$. The result is a short exact sequence with the
middle term free abelian.

Let $\mathbf{N}_{p}$ denote the primitive hull of $H_{2}(D)$ in $H^{2}(M)$ so
that $\mathbf{N}_{p}/H_{2}(D)$ can be identified with the torsion of
$H^{2}(M^{\prime}\smallsetminus D)$. By the universal coefficient theorem, the
torsion subgroup of $H^{2}(M^{\prime}\smallsetminus D)$ is
$\operatorname{Ext}^{1}(G,\mathbb{Z})$ and hence cyclic of order $p$. It
follows that $\mathbf{N}_{p}$ contains $H_{2}(D)$ as a sublattice of index
$p$. The lattice $H_{2}(D)$ is isomorphic to $\mathbf{A}_{p-1}^{\oplus\nu}$.
Since the discriminant group $\mathbf{A}_{p-1}^{\vee}/\mathbf{A}_{p-1}$ is
cyclic of order $p$, the lattice $H_{2}(D)$ (as embedded in
$H^{2}(M^{\prime})$) has discriminant group an $\mathbb{F}_{p}$-vector space
of dimension $\nu$. Since $\mathbf{N}_{p}$ contains $H_{2}(D)$ as a sublattice
of index $p$, the discriminant group $\mathbf{N}_{p}^{\vee}/\mathbf{N}_{p}$ is
an $\mathbb{F}_{p}$-vector space of dimension $\nu-2$. Essentially by
construction we now have an exact sequence

$0\to\mathbf{N}_{p}\to H^{2}(M^{\prime})\to H^{2}(M^{\prime}\smallsetminus
D)_{\textrm{tf}}\to 0,$

where tf stands for the torsion free quotient. The duality between
$H^{2}(M^{\prime}\smallsetminus D)_{\textrm{tf}}$ and
$H_{2}(M^{\prime}\smallsetminus D)$ is realized inside
$H^{2}(M^{\prime};\mathbb{Q})$: if we regard $H_{2}(M^{\prime}\smallsetminus
D)$ as embedded in $H^{2}(M^{\prime})$ via $H_{2}(M^{\prime}\smallsetminus
D)\to H_{2}(M^{\prime})\cong H^{2}(M^{\prime})$, then it is simply the
orthogonal complement of $\mathbf{N}_{p}$ in $H^{2}(M^{\prime})$ and
$H^{2}(M^{\prime}\smallsetminus D)_{\textrm{tf}}$ is identified with
$H_{2}(M^{\prime}\smallsetminus D)^{\vee}\subset
H_{2}(M^{\prime}\smallsetminus D;\mathbb{Q})\subset
H^{2}(M^{\prime};\mathbb{Q})$ as the image of $H^{2}(M^{\prime})$ under the
orthogonal projection along $\mathbf{N}_{p}$. In particular, this identifies
the discriminant groups of $H_{2}(M^{\prime}\smallsetminus D)$ and
$\mathbf{N}_{p}$ and under this identification the discriminant quadratic
forms are each others opposite. Since the group
$\mathbf{N}_{p}^{\vee}/\mathbf{N}_{p}$ is a $\mathbb{F}_{p}$-vector space,
$pH^{2}(M^{\prime}\smallsetminus D)\subset H_{2}(M^{\prime})$ and hence the
intersection pairing on $H^{2}(M^{\prime}\smallsetminus D)_{\textrm{tf}}$
takes values in $\frac{1}{p}\mathbb{Z}$. In other words, if we multiply the
pairing on $H_{2}(M^{\prime})$ with $p$, we obtain an integral form on
$H^{2}(M^{\prime}\smallsetminus D)_{\textrm{tf}}$: it makes
$H^{2}(M^{\prime}\smallsetminus D)_{\textrm{tf}}(p)$ a lattice.

######  Proposition 4.4.

The map $\pi_{M^{\prime}}^{*}$ identifies
$H^{2}(M^{\prime})/\mathbf{N}_{p}\cong H^{2}(M^{\prime}\smallsetminus
D)_{\textrm{tf}}$ with $H^{2}(M)^{G}$, but multiplies the intersection pairing
with $p$. It also defines an injective morphism of exact sequences of which
the top sequence is naturally split:

${0}$${H_{2}(\tilde{D})}$${H^{2}(\tilde{M})^{G}}$${H^{2}(M)^{G}}$${0}$${0}$${\mathbf{N}_{p}}$${H^{2}(M^{\prime})}$${H^{2}(M^{\prime}\smallsetminus
D)_{\textrm{tf}}}$${0}$cyclecycle$\scriptstyle{\pi_{M^{\prime}}^{*}}$$\scriptstyle{\cong}$

Dually, the map $\pi_{M^{\prime}*}$ induces an isomorphism $H_{2}(M)_{G}\cong
H^{2}(M^{\prime}\smallsetminus D)$ that multiplies the intersection pairing
with $p$. It defines a morphism of exact sequences of which the top sequence
is naturally split:

${0}$${H^{2}(\tilde{D})}$${H_{2}(\tilde{M})_{G}}$${H_{2}(M)_{G}}$${0}$${0}$${\mathbf{N}_{p}^{\vee}}$${H_{2}(M^{\prime})}$${H_{2}(M^{\prime}\smallsetminus
D)}$${0}$$\scriptstyle{\pi_{M^{\prime}*}}$$\scriptstyle{\cong}$

###### Proof.

We first examine the unramified $G$-cover $M\smallsetminus M^{G}\to
M^{\prime}\smallsetminus D$. For such a cover we have a spectral sequence

$E^{p,q}_{2}=H^{p}(G;H^{q}(M\smallsetminus M^{G}))\Rightarrow
H^{p+q}(M^{\prime}\smallsetminus D).$

Since $M^{G}$ is finite and $M$ is a simply connected 4-manifold,
$H^{1}(M\smallsetminus M^{G})=0$ and $H^{2}(M)\to H^{2}(M\smallsetminus
M^{G})$ is an isomorphism. So $E^{p,1}_{2}=0$. On the other hand,
$E^{p,0}_{2}=H^{p}(G;\mathbb{Z})$ is trivial in odd degrees and a copy of
$\mathbb{F}_{p}$ in positive even degrees, whereas
$E^{0,q}_{2}=H^{q}(M\smallsetminus M^{G})^{G}$. It follows that
$E^{p,q}_{\infty}=E^{p,q}_{2}$ when $p+q=2$ so that we have a short exact
sequence

$0\to\mathbb{F}_{p}\to H^{2}(M^{\prime}\smallsetminus D)\to H^{2}(M)^{G}\to
0.$ (4.2)

This establishes the isomorphism $H^{2}(M^{\prime}\smallsetminus
D)_{\textrm{tf}}\cong H^{2}(M)^{G}$. The isomorphism $H_{2}(M)_{G}\cong
H^{2}(M^{\prime}\smallsetminus D)$ is obtained in a similar manner by means of
the spectral sequence

$E_{p,q}^{2}=H_{p}(G;H_{q}(M\smallsetminus M^{G}))\Rightarrow
H_{p+q}(M^{\prime}\smallsetminus D),$

where we now need to observe that again $E_{p,1}^{2}=0$, but that
$E_{p,0}^{2}=H_{p}(G;\mathbb{Z})$ is a copy of $\mathbb{F}_{p}$ in odd
positive degrees and trivial in even positive degrees.

The lower exact sequence of the first diagram maps to the exact sequence of
the pair $(\tilde{M},\tilde{M}\smallsetminus\tilde{D})$, of which the relevant
part is

$H^{1}(M\smallsetminus M^{G})\to H_{2}(\tilde{D})\to H^{2}(\tilde{M})\to
H^{2}(M\smallsetminus M^{G})\to H_{1}(\tilde{D}).$

The extremal terms vanish: we already noted that $H^{1}(M\smallsetminus
M^{G})=0$ and $H^{2}(M\smallsetminus M^{G})\cong H^{2}(M)$ and since the
connected components of $\tilde{D}$ are simply connected,
$H_{1}(\tilde{D})=0$. This not only gives a short exact sequence and also
identifies it (via Poincaré duality on $M$ and $\tilde{M}$) with the one
associated to the pair $(\tilde{M},\tilde{D})$:

$0\to H_{2}(\tilde{D})\to H_{2}(\tilde{M})\xrightarrow{\pi_{M*}}H_{2}(M)\to
0,$

and this sequence is split by $\pi_{M}^{*}$. It therefore remains exact if we
pass to $G$-invariants. This proves the exactness of the top sequence.

Since $\pi_{M^{\prime}_{*}}$ takes the fundamental class of $\tilde{M}$ to $p$
times the fundamental class of $M^{\prime}$, the vertical maps multiply the
intersection forms with $p$. The splitting property follows from the fact that
$\tilde{M}\to M$ is an iterated blow-up: the image of the cycle map
$H_{2}(\tilde{D})\to H^{2}(\tilde{M})$ supplements the image of $H^{2}(M)\to
H^{2}(\tilde{M})$ and equals its orthogonal complement with respect to the
intersection pairing.

The assertions regarding the second diagram are proved in a similar fashion. ∎

As we will make clear in Subsection 4.3 below, Proposition 4.4 implies that
the Nikulin lattice completely determines the discriminant form of
$\mathbf{L}_{G}$ and hence its ‘genus’.

###  4.3 The Nikulin lattice

In this subsection we give a complete description of $\mathbf{N}_{p}$ as a
lattice. Number the fixed points of $G$ on $M$ by $p_{1},\dots,p_{\nu}$ and
write $D_{i}$ resp. $\tilde{D}_{i}$ for the connected component of $D$ resp.
$\tilde{D}$ over $p_{i}$. We number the irreducible components of $D_{i}$ by
$D_{i,1},\dots,D_{i,p-1}$ so that $D_{i,j}$ and $D_{i,j^{\prime}}$ are
disjoint unless $|j-j^{\prime}|\leq 1$. These irreducible components come with
an orientation and we will identify them with the homology classes in
$H_{2}(D)$ that they represent. Thus $H_{2}(D_{i})$ is as a lattice isomorphic
with $\mathbf{A}_{p-1}(-1)$ and the basis $\\{D_{i,j}\\}$ of $H_{2}(D)$
consists of $(-2)$-vectors and makes up a system of simple roots.

The group $H_{2}(D_{i})^{\vee}/H_{2}(D_{i})$ is cyclic of order $p$. The way
we numbered the basis of $H_{2}(D_{i})$ yields a natural representative for a
generator, namely $\varpi_{i}:=\frac{1}{p}\sum_{j=1}^{p-1}jD_{i,j}$. This is
in fact the antipode of a fundamental weight: $\varpi_{i}\cdot D_{i,j}=0$
unless $j=p-1$, in which case we get $-1$. This also shows that
$\varpi_{i}\cdot\varpi_{i}=\frac{1-p}{p}$. Note however that if we number the
elements of $D_{i}$ in the opposite direction (replacing $D_{i,j}$ by
$D_{i,p-j}$), then this replaces $j$ by $p-j$.

A generator of $\mathbf{N}_{p}/H_{2}(D)$ can be represented by some $\varpi\in
H_{2}(D)^{\vee}$. We write this as $\varpi:=\sum_{i=1}^{\nu}k_{i}\varpi_{i}$
with $k_{i}\in\\{0,1,\dots,p-1\\}$. If we replace in the exact sequence above
$M^{\prime}$ by a regular neighborhood $U_{i}$ of $D_{i}$, then the preimage
of $U_{i}$ will still be connected and so the same argument shows that
$k_{i}\not=0$. But since the numbering of the elements of $D_{i}$ in the
opposite direction replaces $D_{i,j}$ by $D_{i,p-j}$ (and hence $k_{i}$ by
$p-k_{i}$), only the image of $k_{i}^{2}$ in $\mathbb{F}_{p}^{\times}$ is
well-defined. The following proposition not only determines these images, but
also gives a concrete geometric interpretation of them.

######  Proposition 4.5.

After possible renumbering, the image of $(k_{1}^{2},\dots,k_{\nu}^{2})$ in
$(\mathbb{F}_{p}^{\times})^{\nu}$ is for $p=2,3,5,7$ equal to respectively
$(1,1,1,1,1,1,1,1)$, $(1,1,1,1,1,1)$, $(1,1,4,4)$ and $(1,4,2)$. If we then
let $(k_{1},\dots,k_{\nu})$ equal respectively $(1,1,1,1,1,1,1,1)$,
$(1,1,1,1,1,1)$, $(1,1,2,2)$ and $(1,2,3)$ and let

$\varpi:=\sum_{i=1}^{\nu}\frac{k_{i}}{p}\sum_{j=1}^{p-1}jD_{i,j},$

then $\mathbf{N}_{p}=H_{2}(D)+\mathbb{Z}\varpi$ is an even overlattice of
$H_{2}(D)$ whose $(-2)$-vectors are those of $H_{2}(D)$. Furthermore, if $g\in
G$ has characters $\chi_{i},\chi_{i}^{-1}$ in $T_{p_{i}}M$ (for a complex
structure on $T_{p_{i}}M$ compatible with the given orientation), then
$\\{\chi_{i},\chi_{i}^{-1}\\}=\\{\chi_{1}^{k_{i}},\chi_{i}^{-k_{i}}\\}$.

This completely determines the isomorphism types of the lattices
$H_{2}(M^{\prime}\smallsetminus D)$ and $\mathbf{\Lambda}^{G}$.

###### Proof.

Since $\varpi\in\mathbf{N}_{p}$, we must have

$\frac{1-p}{p}\sum_{i=1}^{\nu}k_{i}^{2}=\varpi\cdot\varpi\in
2\mathbb{Z},\text{ or equivalently,
}\frac{1-p}{2}\sum_{i=1}^{\nu}k_{i}^{2}\equiv 0\pmod{p}.$

If $p=2$ then there is nothing to verify, except perhaps to observe that
$\frac{-1}{2}\cdot 8$ is indeed even. If $p$ is odd then
$\frac{1-p}{2}\in\mathbb{F}_{p}^{\times}$ and so this is then equivalent to
$\sum_{i=1}k_{i}^{2}\equiv 0\pmod{p}$. The squares in
$\mathbb{F}_{3}^{\times}$, $\mathbb{F}_{5}^{\times}$,
$\mathbb{F}_{7}^{\times}$ are resp. $\\{1\\}$, $\\{1,4\\}$, $\\{1,2,4\\}$. The
linear combinations that are divisible by $p$ admit a unique solution: up to a
permutation the image of $(k_{1}^{2},\dots,k^{2}_{\nu})$ in
$(\mathbb{F}_{p}^{\times})^{\nu}$ is equal to resp. $(1,1,1,1,1,1)$,
$(1,1,4,4)$ and $(1,2,4)$. This clearly fixes the isomorphism type of
$\mathbf{N}_{p}$. For the given choices of $(k_{1},\dots,k_{\nu})$ we see that
$\varpi\cdot\varpi$ is resp. $-4$, $-4$, $-8$, $-12$, and one can check that
this does not introduce any new $(-2)$-vectors in $\mathbf{N}_{p}$.

For the assertion regarding the action of $G$ on the tangent spaces at the
fixed points, let $W_{i}$ be the ball-like neighborhood of $p_{i}$ that we
used to complex-linearize the $G$-action at $p_{i}$. So we obtain a regular
neighborhood $W^{\prime}_{i}$ of $D_{i}$ in $M^{\prime}$ for which
$W^{\prime}_{i}\smallsetminus D_{i}\subset M^{\prime}\smallsetminus D$ induces
an isomorphism on fundamental groups and hence defines an isomorphism $G\cong
H_{1}(W^{\prime}_{i}\smallsetminus D_{i})$. The exact sequences for the pair
$(W^{\prime}_{i},W^{\prime}_{i}\smallsetminus D_{i})$ and $(M,M\smallsetminus
D)$ produce the commutative diagram

${H_{2}(M)}$${H^{2}(D)}$${H_{1}(M\smallsetminus
D)}$${0}$${H_{2}(D_{i})}$${H^{2}(D_{i})}$${H_{1}(W^{\prime}_{i}\smallsetminus
D_{i})}$${0}$$\scriptstyle{\cong}$

in which the middle map is given by projection. The upper sequence identifies
$H_{1}(M\smallsetminus D)$ with the cokernel of $H_{2}(M^{\prime})\to
H^{2}(D)$, which is just $H_{2}(D)^{\vee}/\mathbf{N}_{p}^{\vee}$. The element
$\varpi\in\mathbf{N}_{p}$ defines a map $x\in H_{2}(D)^{\vee}\mapsto
x\cdot\varpi\in\mathbb{Q}$ takes values in $\frac{1}{p}\mathbb{Z}$ and is of
course $\mathbb{Z}$-valued on $\mathbf{N}_{p}^{\vee}$. Indeed, its reduction
modulo $\mathbb{Z}$ defines an isomorphism
$H_{2}(D)^{\vee}/\mathbf{N}_{p}^{\vee}\cong\frac{1}{p}\mathbb{Z}/\mathbb{Z}$,
so that $x\mapsto\exp(2\pi\sqrt{-1}x\cdot\varpi)$ defines a nontrivial
character $\chi$ on $G$. On the other hand, the lower sequence identifies
$H_{1}(W^{\prime}_{i}\smallsetminus D_{i})$ with the cokernel of
$H_{2}(D_{i})\to H^{2}(D_{i})\cong H_{2}(D_{i})^{\vee}$ which has the image of
$\varpi_{i}$ as a generator. Via the identification
$H_{1}(W^{\prime}_{i}\smallsetminus D_{i})\cong G$ the character $\chi$ takes
on this generator the value
$\exp(2\pi\sqrt{-1}\varpi_{i}\cdot\varpi)=\exp(2\pi\sqrt{-1}\varpi_{i}\cdot
k_{i}\varpi_{i})=\exp(2\pi\sqrt{-1}k_{i}/p)$, and so the assertion follows.

The discriminant group of $H_{2}(M^{\prime}\smallsetminus D)$ is canonically
isomorphic to the one of $\mathbf{N}_{p}$ (and so is an
$\mathbb{F}_{p}$-vector space of dimension $\nu-2$) and under this isomorphism
their discriminant quadratic forms are opposite. By Proposition 4.4,
$H_{2}(M^{\prime}\smallsetminus D)$ has rank $2\nu-2$ and its intersection
pairing is indefinite. Since that rank is at least two more than the number
($\nu-2$) of generators of the discriminant group of
$H_{2}(M^{\prime}\smallsetminus D)$, this completely determines the
isomorphism type of $H_{2}(M^{\prime}\smallsetminus D)$ by Corollary 1.13.3 of
[Nik2]. This is then also true for $\mathbf{\Lambda}^{G}$ since

$\mathbf{\Lambda}^{G}\cong H^{2}(M^{\prime}\smallsetminus
D)_{\textrm{tf}}(p)=H_{2}(M^{\prime}\smallsetminus D)^{\vee}(p)$

by Proposition 4.4. ∎

######  Remark 4.6.

The numbers $k_{i}$ in Proposition 4.5 are recognized in terms of an elliptic
fibration (see Figure 1 and Subsection 4.7 for more details) as the rotation
numbers: we can label the $I_{p}$-fibers $F_{1},\dots,F_{\nu}$ such that $g\in
G$ rotates in $F_{i}$ over $k_{i}$ times the angle over which it rotates in
$F_{1}$.

######  Lemma 4.7.

The lattice $\mathbf{N}_{p}$ has the following properties.

  1. 1.

The $(-2)$-vectors of $\mathbf{N}_{p}$ are those of $H_{2}(D)$; that is, the
elements of the root system $R=\sqcup_{i=1}^{\nu}R_{i}$, with each $R_{i}$ of
type $A_{p-1}$ with root basis $\\{D_{i,j}\\}_{j}$.

  2. 2.

The discriminant group of $\mathbf{N}_{p}$ is represented by the set of
$\mathbb{F}_{p}$-linear combinations $\sum_{i=1}^{\nu}l_{i}\varpi_{i}$ for
which $\sum_{i}k_{i}l_{i}=0$ modulo the span of
$\varpi=\sum_{i}k_{i}\varpi_{i}$.

  3. 3.

The group of orthogonal transformations of $\mathbf{N}_{p}$ that act trivially
on its discriminant group $\mathbf{N}_{p}^{\vee}/\mathbf{N}_{p}$ is the Weyl
group $W(R)=\prod_{i}W(R_{i})$ of $R$.

###### Proof.

For $(k_{1},\dots,k_{\nu})$ as above, we see that
$\varpi\cdot\varpi=(1-p)/p.\sum_{i}k_{i}^{2}$ is resp. $-4$, $-4$, $-8$, $-12$
and one may check that this does not introduce any new $(-2)$-vectors in
$\mathbf{N}_{p}$. This shows that $R$ is as claimed. As to the other
statement, we shall assume that $p$ is odd, leaving the case $p=2$ as an
exercise (which amounts to a minor modification of the proof below).

The orthogonal group of $\mathbf{N}_{p}$ is a semi-direct product of the Weyl
group $W(R)$ and a permutation group of the collection $\\{D_{i,j}\\}_{i,j}$.
This permutation group has as a subgroup the permutations which preserve each
irreducible component. That subgroup can then also be represented by the
transformations which induce in each component plus or minus the identity
(minus the identity is not in a Weyl group of type $A_{p-1}$ if $p>2$). Hence
we can realize this group also as one which permutes the $\varpi_{i}$’s up to
sign. Consider the inclusions

$H_{2}(D)\subset\mathbf{N}_{p}\subset\mathbf{N}_{p}^{\vee}\subset
H_{2}(D)^{\vee}.$

The discriminant group $H_{2}(D)^{\vee}/H^{2}(D)$ is an
$\mathbb{F}_{p}$-vector space with basis $\\{\varpi_{i}\\}_{i=1}^{\nu}$ and
$\varpi=\sum_{i}k_{i}\varpi_{i}$ defines an isotropic vector in it that
defines a generator of $\mathbf{N}_{p}/H_{2}(D)$. Since
$\varpi_{i}\cdot\varpi_{i}=(1-p)/p$, it follows that
$\mathbf{N}_{p}^{\vee}/H_{2}(D)$ is represented in terms of this basis by the
hyperplane of $l=(l_{1},\dots l_{\nu})\in\mathbb{F}_{p}^{\nu}$ with
$\sum_{i}l_{i}k_{i}=0$. So $\mathbf{N}_{p}^{\vee}/\mathbf{N}_{p}$ is
identified with the quotient of this hyperplane by the span of
$k=(k_{1},\dots,k_{\nu})$. The group of permutations of the basis vectors of
$\mathbb{F}_{p}^{\nu}$ up to sign acts faithfully on this quotient and thus
the lemma follows. ∎

We now describe our candidate for $\mathbf{L}_{G}$ in terms of the Nikulin
lattice $\mathbf{N}_{p}$. For this, let $\varrho\in H_{2}(D)^{\vee}$ be the
Weyl vector, characterized by the property that it takes the value $1$ on each
$D_{i,j}$. So if we write this out on the basis $\\{D_{i,j}\\}_{i,j}$, then

$\varrho=\sum_{i=1}^{\nu}\varrho_{i}\text{ with
}\varrho_{i}=-\tfrac{1}{2}\sum_{j=1}^{p-1}j\cdot(p-j)D_{i,j}.$

When $p$ is odd, this is in fact an element of $H_{2}(D)$. We compute

$\varrho\cdot\varrho=\sum_{i=1}^{\nu}\varrho_{i}\cdot\varrho_{i}=-\frac{\nu}{2}\sum_{j=1}^{p-1}j\cdot(p-j)=-\frac{\nu(p-1)p(p+1)}{2\cdot
6}=-2(p-1)p,$ (4.3)

where we used that $\nu(p+1)=24$. In particular, $\varrho\cdot\varrho\equiv
0\pmod{p}$.

We note that $\varrho_{i}\cdot\varpi_{i}$ is minus the coefficient of
$D_{i,p-1}$ in $\varrho_{i}$, which is $-\frac{1}{2}(p-1)$, and so

$\textstyle\varrho\cdot\varpi=\sum_{i=1}^{\nu}\varrho_{i}\cdot
k_{i}\varpi_{i}=\frac{1}{2}(p-1)\sum_{i=1}^{\nu}k_{i}.$ (4.4)

###  4.4 A Leech type sublattice of the Nikulin lattice

The following lattice will play a central role in the rest of this paper.

######  Definition 4.8 (The lattice $\mathbf{L}_{p}$).

Let $\mathbf{L}_{p}\subset\mathbf{N}_{p}$ to be the sublattice of the vectors
$x\in\mathbf{N}_{p}$ on which $\varrho\cdot D_{i,j}\equiv 0\pmod{p}$ for all
$i,j$.

Note that by the above computation $\varrho$ lies in $\mathbf{L}_{p}$. We will
see that $\mathbf{L}_{p}$ is a lattice which possesses some remarkable
properties. To illustrate this, let us observe that for $p=2$ this reproduces
the lattice $\mathbf{E}_{8}(-2)$. Indeed, $\mathbf{L}_{2}(-\frac{1}{2})$ is by
definition the set of $(x_{1},\dots,x_{8})\in(\frac{1}{2}\mathbb{Z})^{8}$ for
which the each difference $x_{i}-x_{j}$ is integral and $\sum_{i}x_{i}$ is
even and this is one of the classical descriptions of $\mathbf{E}_{8}$
[Serre].

The Weyl group $W(R_{i})$ contains an element $\sigma_{i}$ of order $p$ that
takes $D_{i,j}$ to $D_{i,j+1}$ for $j=1,\dots,p-2$, and $D_{i,p-1}$ to
$D_{i,0}:=-\sum_{j=1}^{p-1}D_{i,j}$, and $D_{i,0}$ back to $D_{i,1}$ (it is a
Coxeter element). Let
$\sigma=(\sigma_{1}^{k_{1}},\dots,\sigma_{\nu}^{k_{\nu}})\in W(R)$. Any
element of $W(R)$, so in particular $\sigma$, leaves invariant any overlattice
of $H_{2}(D)$ such as $\mathbf{N}_{p}$.

######  Proposition 4.9.

The sublattice $\mathbf{L}_{p}$ of $\mathbf{N}_{p}$ is of index $p$, and
contains $\varrho$. It has also the following properties

  1. 1.

$\mathbf{L}_{p}$ is negative definite of rank $\nu(p-1)$ and contains no
$(-2)$-vector(444In the terminology of Nikulin, $\mathbf{L}_{p}$ is a lattice
of Leech-type.).

  2. 2.

The discriminant group $\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$ is an
$\mathbb{F}_{p}$-vector space of dimension $\nu$.

  3. 3.

The group $C_{p}$ of automorphisms of $\mathbf{L}_{p}$ that act trivially on
$\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$ is cyclic of order $p$ (and generated
by $\sigma$).

###### Proof.

The proof is in three steps.

_Step 1: The sublattice $\mathbf{L}_{p}$ of $\mathbf{N}_{p}$ is of index $p$,
contains $\varrho$, but no $(-2)$-vector and
$\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$ is an $\mathbb{F}_{p}$-vector space of
dimension $\nu$._

It is clear that $x\in\mathbf{N}_{p}\to\varrho\cdot x\in\mathbb{Z}$ is onto
and so $\mathbf{L}_{p}$ is a sublattice of $\mathbf{N}_{p}$ of index $p$ as
claimed. Hence
$|\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}|=p^{2}\cdot|\mathbf{N}_{p}^{\vee}/\mathbf{N}_{p}|=p^{\nu}$.
The assertion that $\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$ is an
$\mathbb{F}_{p}$-vector space then amounts to the property that
$p\mathbf{L}_{p}^{\vee}\subset\mathbf{L}_{p}$. Since $\frac{1}{p}\varrho$ maps
to a generator of $\mathbf{L}_{p}^{\vee}/\mathbf{N}_{p}^{\vee}$, it follows
that $p\mathbf{L}_{p}^{\vee}=p\mathbf{N}_{p}^{\vee}+\mathbb{Z}\varrho$. The
latter is a subgroup of $\mathbf{N}_{p}$ on which $\varrho$ takes values in
$\mathbb{Z}p$, hence is contained in $\mathbf{L}_{p}$.

We now check that $\mathbf{L}_{p}$ contains no $(-2)$-vectors. Since we
already know that all the $(-2)$-vectors in $\mathbf{N}_{p}$ lie in
$H_{2}(D)$, it suffices to show that $\varrho$ takes on every root in
$H_{2}(D)$ a value that is not divisible by $p$. To this end, we note that
every positive root in $H_{2}(D_{i})$ is of the form $\sum_{j=r}^{s}D_{i,j}$
with $1\leq r<s\leq p-1$; on such a root $\varrho$ takes the value $r-s$ and
this is never divisible by $p$.

_Step 2: $\sigma$ preserves $\mathbf{L}_{p}$ and acts trivially on
$\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$._

We must show that $\sigma-1$ takes $\mathbf{L}_{p}^{\vee}$ to
$\mathbf{L}_{p}$. We again use that
$\mathbf{L}_{p}^{\vee}=\mathbf{N}_{p}^{\vee}+\frac{1}{p}\mathbb{Z}\varrho$ and
that $\mathbf{N}_{p}^{\vee}/H_{2}(D)$ is represented by the
$\mathbb{Z}$-linear combinations $\sum_{i}l_{i}\varpi_{i}$ for which
$\sum_{i}k_{i}l_{i}\equiv 0\pmod{p}$. We therefore first check this on
$H_{2}(D)$ (regarded as a sublattice of $\mathbf{N}_{p}^{\vee}$).

For this we note that $\sigma(D_{i,j})-D_{i,j}$ (with $j=1,\dots,p-1$) is of
the form $D_{i,j^{\prime}}-D_{i,j}$ where $j^{\prime}$ can also take the value
$0$. Its inner product with $\varrho$ is therefore zero or $-p$, so that this
is an element of $\mathbf{L}_{p}$.

Next we check this for $\sum_{i}l_{i}\varpi_{i}$ with
$\sum_{i}k_{i}l_{i}\equiv 0\pmod{p}$. We compute that

$\sigma_{i}^{k_{i}}(\varpi_{i})=\varpi_{i}-\sum_{j=k_{i}}^{p-1}D_{i,j}$

and so

$\sigma(\sum_{i}l_{i}\varpi_{i})-\sum_{i}l_{i}\varpi_{i}=-\sum_{i=1}^{\nu}l_{i}\sum_{j=k_{i}}^{p-1}D_{i,j}.$

To test whether the right hand side lies in $\mathbf{L}_{p}$, we take its
inner product with $\varrho$. This gives
$-\sum_{i}l_{i}(p-k_{i})=\sum_{i}k_{i}l_{i}-p\sum_{i}l_{i}$ and this is indeed
a multiple of $p$. This proves that $\sigma-1$ takes $\mathbf{N}_{p}^{\vee}$
to $\mathbf{L}_{p}$.

We finally show that $(\sigma-1)(\frac{1}{p}\varrho)\in\mathbf{L}_{p}$. A
straightforward computation shows that

$\textstyle(\sigma-1)(\frac{1}{p}\varrho_{i})=(\sigma_{i}^{k_{i}}-1)(\frac{1}{p}\varrho_{i})=k_{i}\varpi_{i}-\sum_{j=1}^{k_{i}}jD_{i,p-j}$

so that

$\textstyle(\sigma-1)(\frac{1}{p}\varrho)=\varpi-\sum_{i=1}^{\nu}\sum_{j=1}^{k_{i}}jD_{i,p-j}.$

Using Equation 4.4 we see that the value of $\varrho$ on the right-hand side
is

$\sum_{i=1}^{\nu}\tfrac{1}{2}k_{i}(p-1)-\sum_{i=1}^{\nu}\tfrac{1}{2}k_{i}(k_{i}-1)=-\tfrac{1}{2}\sum_{i=1}^{\nu}k_{i}(k_{i}-p).$
(4.5)

If $p=2$ then the right-hand side of (4.5) is $-\tfrac{1}{2}\cdot 8=-4$, which
is indeed even; if $p$ is odd then the fact that $\sum_{i}k_{i}^{2}$ is
divisible by $p$ implies that the right-hand side of (4.5) is also divisible
by $p$. So in all cases, $(\sigma-1)(\frac{1}{p}\varrho)\in\mathbf{L}_{p}$.

_Step 3: Any automorphism of $\mathbf{L}_{p}$ that acts trivially on
$\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$ is a power of $\sigma$._

First note that the $W(R_{i})$-orbit of $\varpi_{i}$ is the $p$-element set
$\\{\varpi_{i}-\sum_{j=p-r}^{p-1}D_{i,j}\\}_{r=0}^{p-1}$ and that this
realizes $W(R_{i})$ as the full permutation group of that orbit. By taking the
inner product with $\varrho_{i}$ and then reducing modulo $p$, this orbit gets
identified with $\mathbb{F}_{p}$. We thus faithfully represent $W(R)$ as a
permutation representation of $\mathbb{F}_{p}^{\nu}$.

Let $g\in\operatorname{O}(\mathbf{L}_{p})$ act trivially on
$\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$. Since $\mathbf{N}_{p}$ is an
overlattice of $\mathbf{L}_{p}$, it then follows that $g$ preserves
$\mathbf{N}_{p}$ and acts trivially on $\mathbf{N}_{p}^{\vee}/\mathbf{N}_{p}$.
According to Lemma 4.7, $g$ will then be an element of the Weyl group $W(R)$.
This element must preserve $H_{2}(D_{i})\cap\mathbf{L}_{p}$. Now an element of
$R_{i}$ (the set $(-2)$-vectors in $H_{2}(D_{i})$) is up to sign of the form
$\sum_{j=r}^{s}D_{i,j}$ for some $1\leq r<s\leq p-1$ and on that vector
$\varrho$ takes the value $r-s$. This shows that the elements of $R_{i}$ on
which $\varrho$ takes a value $1\pmod{p}$ is $\\{D_{i,j}\\}_{j=0}^{p-1}$. It
is clear that $g$ must permute these roots. Notice that the “Dynkin diagram”
of $\\{D_{i,j}\\}_{j=0}^{p-1}$ (given by their inner products) is that of a
$p$-gon. Hence $g$ acts on this $p$-gon as a rotation: it cannot reverse its
orientation because such a transformation will not be in the Weyl group
$W(R_{i})$ and so it will be a rotation. This means that $g$ acts in
$H_{2}(D_{i})$ as some power $\sigma^{n_{i}}_{i}$ of the Coxeter
transformation $\sigma_{i}$.

Let $l=(l_{1},\dots,l_{\nu})\in\mathbb{Z}^{\nu}$ be such that
$\sum_{i}k_{i}l_{i}\equiv 0\pmod{p}$, so that
$\varpi(l):=\sum_{i}l_{i}\varpi_{i}\in\mathbf{L}_{p}^{\vee}$. We are given
that $(g-1)\varpi(l)\in\mathbf{L}_{p}$. In particular,
$\rho\cdot\big{(}(g-1)\varpi(l)\big{)}\equiv 0\pmod{p}$. Now
$g(\varpi_{i})-\varpi_{i}=\sigma_{i}^{n_{i}}(\varpi_{i})-\varpi_{i}=-\sum_{j=n_{i}}^{p-1}D_{i,j}$
and so

$0\equiv\rho\cdot\big{(}(g-1)\varpi(l))=\sum_{i=1}^{\nu}l_{i}\rho_{i}\cdot(\sum_{j=n_{i}}^{p-1}-D_{i,j})=\sum_{i=1}^{\nu}-l_{i}(p-n_{i})\equiv\sum_{i=1}^{\nu}l_{i}n_{i}\pmod{p}.$

As this must be true for all $l=(l_{1},\dots,l_{\nu})\in\mathbb{Z}^{\nu}$ with
$\sum_{i}l_{i}\equiv 0\pmod{p}$, it follows that $(n_{1},\dots,n_{\nu})$ is
proportional to $(k_{1},\dots,k_{\nu})$. This proves that $g=\sigma^{n_{1}}$.
∎

The lattices $\mathbf{L}_{p}$ appear in the literature implicitly beginning
with the work of Nikulin. Although he does not exhibit the lattices
themselves, he proves a uniqueness assertion which we state as follows.

######  Proposition 4.10 (Nikulin [Nik1], Thm. 4.7).

Let $\mathbf{L}$ be a negative-definite lattice without $(-2)$-vectors which
admits an orthogonal automorphism of prime order $p$ that acts trivially on
$\mathbf{L}^{\vee}/\mathbf{L}$. If $\mathbf{L}$ admits an embedding in the K3
lattice $\mathbf{\Lambda}$ then $p\in\\{2,3,5,7\\}$ and
$\mathbf{L}\cong\mathbf{L}_{p}$.

An alternative proof of Theorem 4.10 using Niemeier lattices was later given
by Hashimoto [Ha]. The lattices themselves have been exhibited by Morrison
[Mor] and Van Geemen-Sarti [vGS] for $p=2$ and by Garbagnati-Sarti [GS] for
the odd primes in a case-by-case manner (they denote the lattice
$\mathbf{L}_{p}$ by $\Omega_{p}$).

The $\mathbf{L}_{p}$ comes with the structure of a rank $\nu$ module over
$\mathbb{Z}[C_{p}]$. Since the co-augmentation acts trivially, this factors
through a $\mathbb{Z}(C_{p})$-module structure. The $C_{p}$-invariance of the
symmetric bilinear form on $\mathbf{L}_{p}$ implies that the map
$h:\mathbf{L}_{p}\times\mathbf{L}_{p}\to\mathbb{Z}[C_{p}]$ defined by

$h(u,u^{\prime}):=\sum_{g\in C_{p}}(u\cdot gu^{\prime})g$

is a hermitian form, by which we mean that $h$ is $\mathbb{Z}[C_{p}]$-linear
in the first variable and that $\overline{h(u,u^{\prime})}=h(u^{\prime},u)$,
where the bar is the anti-involution in $\mathbb{Z}[C_{p}]$ induced by the
inversion $g\in C_{p}\mapsto g^{-1}\in C_{p}$. If we compose $h$ with the
augmentation we get $\sum_{g\in C_{p}}(u\cdot gu^{\prime})=0$ and so $h$ takes
its values in the augmentation ideal. For that reason, we may as well compose
$h$ with the projection $\mathbb{Z}[C_{p}]\to\mathbb{Z}(C_{p})$. We thus
recover for odd $p$ (up to scalar) the hermitian modules found by Garbagnati-
Sarti [GS] (they also identify $\mathbf{L}_{3}$ as the Coxeter-Todd lattice).
A priori the group $\operatorname{O}(\mathbf{L}_{p})$ will for odd $p$ only
induce semi-linear transformations for this structure (meaning that they can
induce a nontrivial Galois automorphism of $\mathbb{Z}(C_{p})$); these will
all be linear precisely when $C_{p}$ is central in
$\operatorname{O}(\mathbf{L}_{p})$.

Theorem 4.1 of [GS] states that

$p=3$

implies $\mathbf{\Lambda}^{G}\cong\mathbf{U}\oplus\mathbf{U}(3)^{\oplus
2}\oplus\mathbf{A}_{2}(-1)^{\oplus 2}$,

$p=5$

implies $\mathbf{\Lambda}^{G}\cong\mathbf{U}\oplus\mathbf{U}(5)^{\oplus 2}$,

$p=7$

implies
$\mathbf{\Lambda}^{G}\cong\mathbf{U}(7)\oplus(\begin{smallmatrix}2&1\\\
1&4\end{smallmatrix})$ (the second summand can be identified with the
orthogonal complement of an embedding of $\mathbf{A}_{6}$ in
$\mathbf{E}_{8}$).

######  Remark 4.11.

Recall that the $\nu$-dimensional $\mathbb{F}_{p}$-vector space
$\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$ comes with a nondegenerate quadratic
form. There is an evident homomorphism
$\operatorname{O}(\mathbf{L}_{p})\to\operatorname{O}(\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p})$,
which by Proposition 4.9 has as kernel the group generated by $\sigma$. We
will denote this group by $C_{p}$. For $p=2$ this map is onto: we already
noticed that $\mathbf{L}_{2}\cong\mathbf{E}_{8}(-2)$ so that
$\mathbf{L}_{2}^{\vee}/\mathbf{L}_{2}\cong\mathbf{E}_{8}\otimes\mathbb{F}_{2}$,
and it is well-known that the orthogonal group of the latter is the Weyl group
of $\mathbf{E}_{8}$ modulo its center (see for example Bourbaki, _Groupes et
Algèbres de Lie_ , Ch. 6, Exercice §4, 1). This also implies that the
discriminant form on $\mathbf{L}_{2}$ is different than that of
$\mathbf{A}_{1}(-2)^{\oplus 8}$. On the other hand, one can show that for $p$
odd, $\mathbf{L}_{p}$ has the same genus as $\mathbf{A}_{p-1}^{\oplus\nu}$.
This is one reason why we are not able to push the classification as far as
for $p=2$.

###  4.5 The case $p=2$

Using what we have proved so far in this section, we can now prove Theorem
1.12.

The following corollary illustrates the power of Proposition 4.5.

######  Corollary 4.12.

In case $p=2$, we have $\mathbf{L}_{G}\cong\mathbf{E}_{8}(-2)$. In particular,
$\mathbf{L}_{G}$ contains no $(-2)$-vectors. In fact, $G$ then defines an
involution of Nikulin type in the sense of Definition 1.11: there exists an
isomorphism $\mathbf{\Lambda}\cong\mathbf{U}^{\oplus
3}\oplus\mathbf{E}_{8}(-1)$ via which $G$ acts by exchanging the
$\mathbf{E}_{8}(-1)$-summands.

###### Proof.

We found that the Nikulin lattice $\mathbf{N}_{2}$ has a discriminant group an
$\mathbb{F}_{2}$-vector space of dimension $6$. It is not hard to check that
its discriminant form is that of $\mathbf{U}(2)^{\oplus 3}$. It follows from
Nikulin’s uniqueness theorem that in the situation of Propositions 4.4 and
4.5, the orthogonal complement $H_{2}(M^{\prime}\smallsetminus D)$ of
$\mathbf{N}_{2}$ in $H_{2}(M^{\prime})$ is isomorphic to
$\mathbf{U}(2)^{\oplus 3}\oplus\mathbf{E}_{8}(-1)$. The dual of this lattice
is $\mathbf{U}^{\oplus 3}(\frac{1}{2})\oplus\mathbf{E}_{8}(-1)$ and hence
$\mathbf{\Lambda}^{G}\cong\mathbf{U}^{\oplus 3}\oplus\mathbf{E}_{8}(-2)$ by
Proposition 4.4. This is turn implies that $\mathbf{L}_{G}$ is a lattice whose
discriminant form is that of $\mathbf{E}_{8}(-2)$.

Since $\mathbf{L}_{G}$ has rank $8$ and is negative definite, it follows that
$\mathbf{L}_{G}(\frac{1}{2})$ is unimodular negative definite. It cannot be
odd, because then $\mathbf{L}_{G}\cong\mathbf{A}_{1}(-1)^{\oplus 8}$ and this
has not the discriminant form of $\mathbf{E}_{8}(-2)$. It follows that
$\mathbf{L}_{G}\cong\mathbf{E}_{8}(-2)$ as asserted.

It then also follows that the embedding
$\mathbf{\Lambda}^{G}\oplus\mathbf{L}_{G}\subset\mathbf{\Lambda}$ is
isomorphic to $(\mathbf{U}^{\oplus
3}\oplus\mathbf{E}_{8}(-2))\oplus\mathbf{E}_{8}(-2)\subset\mathbf{U}^{\oplus
3}\oplus\mathbf{E}_{8}(-1)$, where the two copies of $\mathbf{E}_{8}(-2)$
embed in $\mathbf{E}_{8}(-1)^{\oplus 2}$ as resp. the diagonal and the
antidiagonal. It is clear that via such an isomorphism, $G$ simply exchanges
the $\mathbf{E}_{8}(-1)$-summands. ∎

Since we arrived at this Corollary 4.12 via a symplectic resolution, we obtain
additional geometric information, for this also tells us how to recover the
$G$-action topologically: start off with an oriented smooth manifold
$M^{\prime}$ that is homeomorphic to a K3 surface and find in it a disjoint
union $D$ of $8$ embedded $2$-spheres with self-intersection $-2$ such that
$H^{2}(D;\mathbb{Z})$ meets $H^{2}(M^{\prime})$ in
$H^{2}(M^{\prime};\mathbb{Q})$ in $\mathbf{N}_{2}$. This implies that
$M^{\prime}$ admits a double covering $\tilde{M}^{\prime}\to M^{\prime}$
ramified along $D$. The preimage $\tilde{D}$ of $D$ in $\tilde{M}^{\prime}$
consists $8$ pairwise disjoint $2$-spheres with self-intersection $-1$. These
can be contracted to produce a smooth $4$-manifold homeomorphic to the K3
surface. The covering transformation then gives us the $G$-action.

######  Remark 4.13.

This comes close to, but does not quite produce, a classification of
$\operatorname{Diff}(M)$-conjugacy classes of involutions of Nikulin type. In
the complex-analytic setting the resolved orbit space $M^{\prime}$ produces a
genuine K3 surface and one can show that the involutions thus obtained lie in
a single conjugacy class in $\operatorname{Diff}(M)$. It is not clear to us
whether every involution of the type considered here belongs to that class.
For example, we do not know whether our $M^{\prime}$ is always _diffeomorphic_
(rather than homeomorphic) with a K3 surface. But even if this were the case,
then we would still need to know whether the isotopy classes of ‘Nikulin
submanifolds’ $D\subset M^{\prime}$ that define a given primitive embedding of
the Nikulin lattice in $H_{2}(M^{\prime})$ make up a single orbit under the
Torelli group of $M^{\prime}$.

###### Proof of Theorem 1.8.

Let $G\subset\operatorname{Mod}(M)$ be a subgroup of order $2$. Ruberman
(Theorem 2.1 of [Ru]) and Matumoto [Ma2] proved that if a closed spin
$4$-manifold $X$ with $H_{1}(X)=0$ admits a smooth involution acting trivially
on $H_{2}(X)$, then $X$ has signature $0$. Since the K3 manifold $M$ has
$H_{1}(M)=0$ and is spin, but has signature $-16$, it follows that the induced
representation
$G\to\operatorname{Aut}(H_{M})\cong\operatorname{O}^{+}(\mathbf{\Lambda})$ is
faithful. We can therefore identify $G$ with its image in
$\operatorname{O}^{+}(\mathbf{\Lambda})$.

The space $Gr^{+}_{3}(\mathbf{\Lambda}_{\mathbb{R}})$ of positive-definite
3-planes in $\mathbf{\Lambda}_{\mathbb{R}}$ is the symmetric space of
$\operatorname{O}^{+}(\mathbf{\Lambda}_{\mathbb{R}})$, whose natural
$\operatorname{O}^{+}(\mathbf{\Lambda}_{\mathbb{R}})$-metric has nonpositive
sectional curvature. Thus any finite $G$ acting on this space has a fixed
point; that is, $G$ leaves invariant some positive-definite $3$-plane
$P\subset\mathbf{\Lambda}_{\mathbb{R}}$. There are two cases to consider
according to whether or not the induced $G$-action on $P$ is trivial.

If the $G$ action on $P$ is nontrivial then $\mathbf{L}_{G}=0$. Theorem 1.2
then implies that $G$ can be realized as a group of automorphisms of a complex
structure on $M$ endowed with a Ricci-flat Kähler metric. In particular the
subgroup $G\subset\operatorname{Mod}(M)$ can be realized as a group of
diffeomorphisms.

If the $G$-action on $P$ is trivial then Corollary 4.12 implies that a
necessary condition for $G$ to lift to $\operatorname{Diff}(M)$ is that
$\mathbf{L}_{G}$ contain no $(-2)$-vector. But Theorem 1.2 shows that this is
also sufficient: then too $G$ can then be realized as a group of automorphisms
of a complex structure on $M$ endowed with a Ricci-flat Kähler metric. ∎

###  4.6 Nielsen realization relative to a Ricci-flat metric

In this subsection we deduce some geometric consequences of our study of
$\mathbf{L}_{p}$.

######  Lemma 4.14 (Moduli space of Ricci-flat metrics).

Let $i:\mathbf{L}_{p}\hookrightarrow\mathbf{\Lambda}$ be a primitive
embedding. Extend the $C_{p}$ action to $\mathbf{\Lambda}$ by letting it act
trivially on $i(\mathbf{L}_{p})^{\perp}$. Denote by $\mathcal{R}(i)$ the space
of Ricci-flat metrics $s$ on $M$ for which $C_{p}$ lies in the image of the
isometry group of $(M,s)$. Then the orbit space of $\mathcal{R}(i)$ with
respect to the natural action of the Torelli group of $M$ can be identified
with the complement of a locally-finite union of submanifolds of codimension
$3$ in the symmetric space of positive $3$-planes in the orthogonal complement
of $i$ in $\mathbf{\Lambda}_{\mathbb{R}}$; in particular, it is a connected
manifold of dimension $3(2\nu-5)$. This manifold is the base of a fine moduli
space: it supports a universal family of Ricci-flat K3 manifolds.

Conversely, if $s$ is a Ricci-flat metric on $M$, then every cyclic group of
isometries of $(M,s)$ of order $p$ that pointwise fixes a positive $3$-plane
in $\mathbf{\Lambda}_{\mathbb{R}}$ arises in this manner for some primitive
embedding $i:\mathbf{L}_{p}\hookrightarrow\mathbf{\Lambda}$.

###### Proof.

Since $C_{p}$ acts trivially on $\mathbf{L}_{p}^{\vee}/\mathbf{L}_{p}$, its
action on $\mathbf{L}_{p}$ extends to $\mathbf{\Lambda}$ by letting it act as
the identity on the orthogonal complement of $i$; this is the extension to
which our notation $i_{*}C_{p}$ refers. In this way $i_{*}C_{p}$ is a subgroup
of $\operatorname{O}(\mathbf{L}_{p})$.

If $s$ is a Ricci-flat metric on $M$, then in view of the Torelli theorem, the
isometry group of $(M,s)$ acts faithfully on $\mathbf{\Lambda}$ and a
necessary and sufficient condition that this group contains $C_{p}$ is that
the positive $3$-plane $P_{s}\subset\mathbf{\Lambda}_{\mathbb{R}}$ defined by
its self-dual harmonic $2$-forms has the property that its orthogonal
complement $P_{s}^{\perp}$ (so that is the space of antiselfdual harmonic
$2$-forms) does not contain a $(-2)$-vector and contains the image of $i$. In
other words, the moduli space in question is the space of positive $3$-planes
$P\subset i(\mathbf{\Lambda})_{\mathbb{R}}^{\perp}$ for which $P^{\perp}$ does
not contain a $(-2)$-vector. The space of positive $3$-planes $P\subset
i(\mathbf{\Lambda})_{\mathbb{R}}^{\perp}$ is the symmetric space $D_{i}$ of
the orthogonal group of $i(\mathbf{\Lambda})_{\mathbb{R}}^{\perp}$, and the
locus of positive $3$-planes $P\subset
i(\mathbf{\Lambda})_{\mathbb{R}}^{\perp}$ that are also perpendicular to a
given $(-2)$-vector $v$ defines a symmetric subspace of codimension $3$. These
symmetric subspaces are locally finite in $D_{i}$ and hence the moduli space
is as asserted. It is known that this support a universal family (see
Corollary 5.5 of [Lo]).

If $s$ is a Ricci-flat metric on $M$ and $G$ is a subgroup of isometries of
$(M,s)$ of order $p$ which pointwise fixes a positive $3$-plane in
$\mathbf{\Lambda}_{\mathbb{R}}$, then $P_{s}^{\perp}$ contains
$\mathbf{L}_{G}$. In particular, $\mathbf{L}_{G}$ does not contain any
$(-2)$-vector and is therefore by Nikulin’s uniqueness property 4.10 of the
above type. ∎

Building on all of the above material, we can now finish the proof of Theorem
1.6, which classifies all prime order groups $G\subset\operatorname{Diff}(M)$
preserving some complex structure.

###### Proof of Theorem 1.6.

Let us first observe that in the situation of Lemma 4.14 the embedding $i$
determines an orbit of order $p$ subgroups of $\operatorname{Diff}(M)$ with
respect to the action of the Torelli group. This is because we have associated
to $i$ a fine moduli space with connected base, whose monodromy representation
on $\mathbf{\Lambda}$ is trivial.

We now let $i$ vary. The orthogonal complement for every primitive embedding
of $\mathbf{L}_{p}$ in $\mathbf{\Lambda}$ is even and indefinite of rank
$2\nu-2$ (it has signature $(3,2\nu-5)$) and its discriminant group of
$\mathbf{L}_{p}$ has $\nu$ generators. Since $\nu<2\nu-2$, it follows from
Nikulin’s Thm. 1.14.1 and Prop. 1.14.2 [Nik2] that such primitive embeddings
satisfy the weak form of Witt’s theorem, by which is meant that the
sublattices $\mathbf{L}\subset\mathbf{\Lambda}$ isomorphic to $\mathbf{L}_{p}$
make up a single orbit under $\operatorname{O}(\mathbf{\Lambda})$. Since
$\operatorname{O}(\mathbf{\Lambda})=\\{\pm
1\\}\operatorname{O}^{+}(\mathbf{\Lambda})$, it then follows that it is also a
single orbit under $\operatorname{O}^{+}(\mathbf{\Lambda})$. If we combine
this with the preceding, we find that the cyclic subgroups $G$ of
$\operatorname{Diff}(M)$ of order $p$ which preserve a Ricci-flat metric and
for which $L_{G}$ is negative definite make up a single conjugacy class.

For the proof of Theorem 1.6 it remains to make the passage from invariant
Ricci-flat metric metrics to invariant complex structures. As we recalled
earlier [HKLR], the twistor theory shows that a Ricci-flat metric $s$ on $M$
is a Kähler metric for a $2$-sphere of complex structures, and that every
complex structure so arises. In fact, for every complex structure $J$ on $M$
and every automorphism $g$ of $(M,J)$ for which
$\mathbf{L}_{\operatorname{\langle}g\operatorname{\rangle}}$ is negative
definite, there exists a Ricci-flat metric $s$ that is a Kähler metric
relative to $J$ and for which $g$ is an isometry. ∎

###  4.7 Elliptic fibrations

As we saw, if a cyclic group $G$ of prime order acts faithfully on a K3
surface, then $p\in\\{2,3,5,7\\}$. Van Geemen-Sarti (for $p=2$) and
Garbagnati-Sarti (for $p$ odd) show that this can then be realized as a
subgroup of the Mordell-Weil group of an elliptic K3 surface $X$ with
$\nu=24/(p+1)$ fibers of Kodaira type $I_{p}$ and as many fibers of type
$I_{1}$. (We recall here that a $I_{m}$-fiber is for $m>1$ a ‘cycle’ of
$(-2)$-curves, $m$ in number and $I_{1}$ is a rational curve with a node.) So
the sum of the Euler characteristics of the singular fibers is $\nu(p+1)=24$,
which is indeed the Euler characteristic of $M$. The $G$-action is then
realized as the Mordell-Weil group \- the group of automorphisms that preserve
each fiber and act on it as a translation. Each nontrivial element of $G$ acts
in each $I_{p}$-fiber as a nontrivial rotation (but not necessary over the
same angle, see below). It also acts nontrivially in each $I_{1}$-fiber and as
observed earlier, the singular points of such fibers account for all the
$G$-fixed points (see Figure 1).

In what follows we give a homological description of the elliptic fibration
(with and without a section) on which $C_{p}$ acts. The geometrically most
useful one is the case when $C_{p}$ preserves each fiber, but as we will see
for $p\not=7$, there exist also fibrations that permutes the fibers.

Elliptic fibrations with fiberwise $C_{p}$-action. We here describe the
homological model of an elliptic fibration that is fiberwise preserved by the
$C_{p}$-action, as depicted in Figure 1.

Let $\mathbf{Z}$ be a copy of $\mathbb{Z}$, but regarded as a rank one lattice
with zero quadratic form. Denote its generator by $f$. Define an embedding
$i:\mathbf{N}_{p}\to\mathbf{N}_{p}\oplus\tfrac{1}{p}\mathbf{Z}$ by
$x\in\mathbf{N}_{p}\mapsto x+\tfrac{1}{p}(\varrho\cdot x)f$ and let

$\hat{\mathbf{N}}_{p}:=i(\mathbf{N}_{p})+\mathbf{Z}\subset\mathbf{N}_{p}\oplus\tfrac{1}{p}\mathbf{Z}.$

The definition of $\mathbf{L}_{p}$ shows that $\hat{\mathbf{N}}_{p}$ meets the
first summand in $\mathbf{L}_{p}$ and the second summand in $\mathbf{Z}$. For
$i=1,\dots,\nu$ and $j=1,\dots,p-1$ let

$\tilde{D}_{i,j}:=i(D_{i,j})=D_{i,j}+\tfrac{1}{p}f\in\hat{\mathbf{N}}_{p}$