# Involutions on K3 surfaces and derived equivalence

Brendan Hassett  Department of Mathematics
Brown University Box 1917
151 Thayer Street, Providence, RI 02912
USA  brendan<EMAIL_ADDRESS>and  Yuri Tschinkel  Courant Institute
New York University
New York, NY 10012
USA<EMAIL_ADDRESS>Simons Foundation
160 Fifth Avenue
New York, NY 10010
USA

###### Abstract.

We study involutions on K3 surfaces under conjugation by derived equivalence
and more general relations, together with applications to equivariant
birational geometry.

##  1\. Introduction

The structure of $\operatorname{Aut}D^{b}(X)$, the group of autoequivalences
of the bounded derived category $D^{b}(X)$ of a K3 surface $X$, is very rich
but well-understood only when the Picard group $\operatorname{Pic}(X)$ has
rank one [BB17]. The automorphism group $\operatorname{Aut}(X)$ of $X$ lifts
to $\operatorname{Aut}D^{b}(X)$, and one may consider the problem of
classification of finite subgroups $G\subset\operatorname{Aut}(X)$ up to
conjugation – either by automorphisms, derived equivalence, or even larger
groups. This problem is already interesting for cyclic $G$, and even for
involutions, e.g., Enriques or Nikulin involutions. There is an extensive
literature classifying these involutions on a given K3 surface $X$:
topological types, moduli spaces of polarized K3 surfaces with involution, and
the involutions on a single $X$ up to automorphisms, see, e.g., [AN06],
[vGS07], [Oha07], [SV20], [Zha98].

Here we investigate involutions up to derived equivalence, i.e., derived
equivalences respecting involutions. Our interest in “derived” phenomena was
sparked by a result in [Sos10]— there exist complex conjugate, derived
equivalent nonisomorphic K3 surfaces—as well as our investigations of
arithmetic problems on K3 surfaces [HT17], [HT22].

One large class of involutions $\sigma:X\rightarrow X$ are those whose
quotient $Q=X/\sigma$ is rational. Examples include $Q$ a del Pezzo surface
and $X\rightarrow Q$ a double cover branched along a smooth curve
$B\in|-2K_{Q}|$. We may allow $Q$ to have ADE surface singularities away from
$B$, or $B$ to have ADE curve singularities; then we take $X$ as the minimal
resolution of the resulting double cover of $Q$. These were studied by Alexeev
and Nikulin in connection with classification questions concerning singular
del Pezzo surfaces [AN06]. Our principal result here (see Section 5) is that

  * •

equivariant derived equivalences of such $(X,\sigma)$ are in fact equivariant
isomorphisms (see Corollary 11).

Our study of stable equivalence of lattices with involution leads us to a
notion of skew equivalence, presented in Section 6. Here, duality interacts
with the involution which is reflected in a functional equations for the
Fourier-Mukai kernel. Explicit examples, for anti-symplectic actions with
quotients equal to $\mathbb{P}^{2}$, are presented in Section 7.

Next, we focus on Nikulin involutions $\iota:X\to X$, i.e., involutions
preserving the symplectic form, so that the resolution of singularities $Y$ of
the resulting quotient $X/\iota$ is a K3 surface. A detailed study of such
involutions can be found in [vGS07]. In addition to the polarization class,
the Picard group $\operatorname{Pic}(X)$ contains the lattice
$\mathrm{E}_{8}(-2)$; van Geemen and Sarti describe the moduli and the
geometry in the case of minimal Picard rank
$\operatorname{rk}\operatorname{Pic}(X)=9$. In Section 8, we extend their
results to higher ranks, and

  * •

exhibit nontrivial derived equivalences between Nikulin involutions
(Proposition 21).

These, in turn, allow us to construct in Section 9 examples of equivariant
birational isomorphisms $\phi:\mathbb{P}^{4}\dashrightarrow\mathbb{P}^{4}$
with nonvanishing invariant $C_{G}(\phi)$, introduced in [LSZ20], [LS22] and
extended to the equivariant context in [KT22].

The case of Enriques involutions $\epsilon:X\to X$, i.e., fixed-point free
involutions, so that the resulting quotient $X/\epsilon$ is an Enriques
surface, has also received considerable attention. There is a parametrization
of such involutions in terms of the Mukai lattice $\tilde{\mathrm{H}}(X)$, and
an explicit description of conjugacy classes, up to automorphisms
$\operatorname{Aut}(X)$, in interesting special cases, e.g., for K3 surfaces
of Picard rank 11, Kummer surfaces of product type, general Kummer surfaces,
or singular K3 surfaces [Kon92], [Oha07], [Ser05], [SV20]. In Section 10 we
observe that

  * •

the existence of an Enriques involution on a K3 surface $X$ implies that every
derived equivalent surface is equivariantly isomorphic to $X$ (Propositions 28
and 29);

  * •

while there are no nontrivial equivariant derived autoequivalences, we exhibit
nontrivial orientation reversing (i.e., skew) equivalences, e.g., on singular
K3 surfaces.

Acknowledgments: The first author was partially supported by Simons Foundation
Award 546235 and NSF grant 1701659, the second author by NSF grant 2000099. We
are grateful to Nicolas Addington, Sarah Frei, Lisa Marquand, and Barry Mazur
for helpful suggestions.

##  2\. Lattice results

We recall basic terminology and results concerning lattices: torsion-free
finite-rank abelian groups $\mathrm{L}$ together with a nondegenerate integral
quadratic form $(\cdot,\cdot)$, which we assume to be even. Basic examples are

$\mathrm{U}=\left(\begin{matrix}0&1\\\ 1&0\end{matrix}\right)$

and positive definite lattices associated with Dynkin diagrams (denoted by the
same letter).

We write $\mathrm{L}(2)$, when the form is multiplied by $2$. We let

$d(\mathrm{L}):=\mathrm{L}^{*}/\mathrm{L}$

be the discriminant group and

$q_{\mathrm{L}}:d(\mathrm{L})\to\mathbb{Q}/2\mathbb{Z}$

the induced discriminant quadratic form.

### Nikulin’s form of Witt cancellation:

######  Proposition 1.

[Nik79b, Cor. 1.13.4] Given an even lattice $\mathrm{L}$,
$\mathrm{L}\oplus\mathrm{U}$ is the unique lattice with its signature and
discriminant quadratic form.

If lattices $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ are stably isomorphic –
become isomorphic after adding unimodular lattices of the same signature –
then

$\mathrm{L}_{1}\oplus\mathrm{U}\simeq\mathrm{L}_{2}\oplus\mathrm{U}.$

### Nikulin stabilization result:

Given a lattice $\mathrm{L}$, write
$\mathrm{L}_{p}=\mathrm{L}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$ for the induced
$p$-adic bilinear form. The $p$-primary part of $d(\mathrm{L})$ depends only
on $\mathrm{L}_{p}$ and is written $d(\mathrm{L}_{p})$. We use
$q_{\mathrm{L}_{p}}$ for the induced discriminant quadratic form on
$d(\mathrm{L}_{p})$. For a finitely generated abelian group $A$, let $\ell(A)$
denote the minimal number of generators.

######  Proposition 2.

[Nik79b, Thm. 1.14.2] Let $\mathrm{L}$ be an even indefinite lattice
satisfying

  * •

$\operatorname{rank}(\mathrm{L})\geq\ell(d(\mathrm{L}_{p}))+2$ for all $p\neq
2$;

  * •

if $\operatorname{rank}(\mathrm{L})=\ell(d(\mathrm{L}_{2}))$ then
$q_{\mathrm{L}_{2}}$ contains $u^{(2)}_{+}(2)$ or $v^{(2)}_{+}(2)$ as a
summand, i.e., the discriminant quadratic forms of

$\mathrm{U}^{(2)}(2)=\left(\begin{matrix}0&2\\\
2&0\end{matrix}\right),\quad\mathrm{V}^{(2)}(2)=\left(\begin{matrix}4&2\\\
2&4\end{matrix}\right).$

Then the genus of $\mathrm{L}$ admits a unique class and
$\mathrm{O}(\mathrm{L})\rightarrow\mathrm{O}(q_{\mathrm{L}})$ is surjective.

######  Remark 3.

[Nik79b, Rem. 1.14.5] The $2$-adic condition can be achieved whenever the
discriminant group $d(\mathrm{L})$ has $(\mathbb{Z}/2\mathbb{Z})^{3}$ as a
summand.

Thus given a lattice $\mathrm{L}$, any automorphism of
$(d(\mathrm{L}),q_{\mathrm{L}})$ may be achieved via an automorphism of
$\mathrm{L}\oplus\mathrm{U}$. More precisely, given two lattices
$\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ of the same rank and signature and an
isomorphism

$\varrho:(d(\mathrm{L}_{1}),q_{\mathrm{L}_{1}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(d(\mathrm{L}_{2}),q_{\mathrm{L}_{2}})$

there exists an isomorphism

$\rho:\mathrm{L}_{1}\oplus\mathrm{U}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathrm{L}_{2}\oplus\mathrm{U}$

inducing $\varrho$.

### Nikulin imbedding result:

######  Proposition 4.

[Nik79b, Cor. 1.12.3,Thm. 1.14.4] Let $\mathrm{L}$ be an even lattice of
signature $(t_{+},t_{-})$ and discriminant group $d(\mathrm{L})$. Then
$\mathrm{L}$ admits a primitive embedding into a unimodular even lattice of
signature $(\ell_{+},\ell_{-})$ if

  * •

$\ell_{+}-\ell_{-}\equiv 0\mod 8$;

  * •

$\ell_{+}\geq t_{+}$ and $\ell_{-}\geq t_{-}$;

  * •

$\ell_{+}+\ell_{-}-t_{+}-t_{-}>\ell(d(\mathrm{L}))$, the rank of
$d(\mathrm{L})$.

This embedding is unique up to automorphisms if

  * •

$\ell_{+}>t_{+}$ and $\ell_{-}>t_{-}$;

  * •

$\ell_{+}+\ell_{-}-t_{+}-t_{-}\geq 2+\ell(d(\mathrm{L}))$.

In particular, any even nondegenerate lattice of signature $(1,9)$ admits a
unique embedding into the K3 lattice $\mathrm{U}^{\oplus
3}\oplus\mathrm{E}_{8}(-1)^{\oplus 2}.$

##  3\. Mukai lattices and derived automorphisms

Throughout, we work over the complex numbers $\mathbb{C}$. Let $X$ be a
complex K3 surface and

$\operatorname{Pic}(X)\subset\mathrm{H}^{2}(X,\mathbb{Z})\simeq\mathrm{E}_{8}(-1)^{\oplus
2}\oplus\mathrm{U}^{3}$

its Picard lattice, a sublattice of a lattice of signature $(3,19)$, with
respect to the intersection pairing. The Picard lattice determines the
automorphisms of $X$: the natural map

$\operatorname{Aut}(X)\to\mathrm{O}(\operatorname{Pic}(X))/\langle\text{reflections
by $(-2)$-classes}\rangle,$

to the quotient of the orthogonal group of the Picard lattice, has finite
kernel and cokernel. All possible finite $G\subset\operatorname{Aut}(X)$ have
been classified, see [BH21]. Classification of
$\operatorname{Aut}(X)$-conjugacy classes of elements or subgroups boils down
to lattice theory of $\operatorname{Pic}(X)$; we will revisit it in special
cases below.

The transcendental lattice of $X$ is the orthogonal complement

$T(X):=\operatorname{Pic}(X)^{\perp}\subset\mathrm{H}^{2}(X,\mathbb{Z}).$

This lattice plays a special role: two K3 surfacs $X_{1}$, $X_{2}$ are derived
equivalent if and only if there exists an isomorphism of lattices

$T(X_{1})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}T(X_{2}),$

compatible with Hodge structures [Orl97]. Derived equivalence also means that
the lattices $\operatorname{Pic}(X_{1})$ and $\operatorname{Pic}(X_{2})$
belong to the same genus. Over nonclosed fields, or in equivariant contexts,
derived equivalence is a subtle property, see, e.g., [HT17], [HT22].

We recall standard examples of Picard lattices of derived equivalent but not
isomorphic K3 surfaces

######  Remark 5.

In Picard rank one: the number of nonisomorphic derived equivalent surfaces is
governed by the number of prime divisors of the polarization degree $2d$; see
[HLOY04, Cor. 2.7] and Remark 5. The isomorphisms classes correspond to
solutions of the congruence

(3.1) $x^{2}\equiv 1\pmod{4d}$

modulo $\pm 1$. When $d>1$ the number of derived equivalent K3 surfaces is
$2^{\tau(d)-1}$, where $\tau$ is the number of distinct prime factors of $d$.

In Picard rank two: derived equivalences among lattice-polarized K3 surfaces
of square-free discriminant are governed by the genera in the class group of
the corresponding real quadratic field [HLOY04, Sect. 3].

Here are instances where derived equivalence is trivial

######  Proposition 6.

[HLOY04, Cor. 2.6, 2.7] Derived equivalence implies isomorphism in each of the
following cases:

  * •

if the Picard rank is $\geq 12$;

  * •

if the surfaces admits an elliptic fibration with a section;

  * •

if the Picard rank is $\geq 3$ and the discriminant group of the Picard group
is cyclic.

We give a further example in Proposition 21.

Let

$\tilde{\mathrm{H}}(X):=\mathrm{H}^{0}(X,\mathbb{Z})(-1)\oplus\mathrm{H}^{2}(X,\mathbb{Z})\oplus\mathrm{H}^{4}(X,\mathbb{Z})(1)$

be its Mukai lattice, a lattice of signature $(4,20)$, with respect to the
Mukai pairing. There is a surjective homomorphism [HMS09, Cor. 3]

$\operatorname{Aut}D^{b}(X)\to\mathrm{O}^{+}(\tilde{\mathrm{H}}(X))\subset\mathrm{O}(\tilde{\mathrm{H}}(X))$

onto the group of signed Hodge isometries, a subgroup of the orthogonal group
of the Mukai lattice preserving orientations on the positive $4$-planes.

We retain the notation from [HT22, Sect. 2], where we discussed the notion and
basic properties of equivariant derived equivalences between K3 surfaces. We
recall:

> Let $X_{1}$ and $X_{2}$ be K3 surfaces equipped with a generically free
> action of a finite cyclic group $G$. Then $X_{1}$ and $X_{2}$ are
> $G$-equivariantly derived equivalent if and only if there exists a
> $G$-equivariant isomorphism of their Mukai lattices
>
>
> $\tilde{\mathrm{H}}(X_{1})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\tilde{\mathrm{H}}(X_{2})$
>
> respecting the Hodge structures.

Note that the $G$-action is necessarily trivial on

$\mathrm{H}^{0}(X,\mathbb{Z})(-1)\oplus\mathrm{H}^{4}(X,\mathbb{Z})(1).$

Even in the event of an isomorphism $X_{1}\simeq X_{2}$, equivariant derived
equivalences are interesting: indeed, there are actions of finite groups $G$
that are not conjugate in $\operatorname{Aut}(X)$ but are conjugate via
$\operatorname{Aut}D^{b}(X)$ as the action of the latter group is visibly
larger.

Let $G$ be a finite group and $X_{1}$ and $X_{2}$ K3 surfaces with
$G$-actions. For simplicity, assume that $G$ acts on $T(X_{i})$ via
$\pm\mathrm{I}$. (This is the case if the transcendental cohomology is
simple.)

Given a $G$-equivariant isomorphism $T(X_{1})\simeq T(X_{2})$, can we lift to
a $G$-equivariant isomorphism of Mukai lattices

$\tilde{\mathrm{H}}(X_{1},\mathbb{Z})\simeq\tilde{\mathrm{H}}(X_{2},\mathbb{Z}),$

where $G$ acts trivially on the hyperbolic summand

$\mathrm{U}=\mathrm{H}^{0}\oplus\mathrm{H}^{4}?$

Clearly the answer is NO. Suppose that $G=C_{2}=\left<\epsilon\right>$ and the
$\epsilon=-1$ eigenspaces are stably isomorphic but not isomorphic. Adding
$\mathrm{U}$ does nothing to achieve the desired stabilization.

In other words, $\mathrm{U}$ is “too small”. We need to add summands where $G$
acts nontrivially to achieve stabilization across all the various isotypic
components.

##  4\. Cohomological Fourier-Mukai transforms

Let $X_{1}$ and $X_{2}$ be smooth projective complex K3 surfaces. A
fundamental result of Orlov [Orl97] shows that any equivalence

$\Phi:D^{b}(X_{1})\rightarrow D^{b}(X_{2})$

arises from a kernel $\mathcal{K}\in D^{b}(X_{1}\times X_{2})$ through a
Fourier-Mukai transform

$\begin{array}[]{rcl}\Phi_{\mathcal{K}}:D^{b}(X_{1})&\rightarrow&D^{b}(X_{2})\\\
\mathcal{E}&\mapsto&{\pi_{2}}_{*}(\pi_{1}^{*}\mathcal{E}\otimes\mathcal{K}).\end{array}$

All the indicated functors are taken in their derived senses. Given such a
kernel, there is also a Fourier-Mukai transform in the opposite direction

$\begin{array}[]{rcl}\Psi_{\mathcal{K}}:D^{b}(X_{2})&\rightarrow&D^{b}(X_{1})\\\
\mathcal{E}&\mapsto&{\pi_{1}}_{*}(\pi_{2}^{*}\mathcal{E}\otimes\mathcal{K}).\end{array}$

Mukai has computed the kernel of the inverse

$\Phi_{\mathcal{K}}^{-1}=\Psi_{\mathcal{K}^{\vee}[2]}$

i.e., a twist of the dual to our original kernel. See [Muk87, 4.10], [BBHR97,
§ 4.3], and [Huy06, p. 133] for details. The computation relies on
Grothendieck-Serre Duality, so the appearance of the dualizing complex is
natural. This machinery [Huy06, § 3.4] also allows us to analyze how Fourier-
Mukai transforms interact with taking duals:

$\displaystyle\Phi_{\mathcal{K}}(\mathcal{E}^{\vee})$
$\displaystyle={\pi_{2}}_{*}(\mathcal{K}\otimes\pi_{1}^{*}(\mathcal{E}^{\vee}))$
$\displaystyle=(({\pi_{2}}_{*}(\mathcal{K}^{\vee}\otimes\pi_{1}^{*}\mathcal{E}))^{\vee})[-2]$
$\displaystyle=((\Phi_{\mathcal{K}^{\vee}}\mathcal{E})[2])^{\vee}$
$\displaystyle=(\Phi_{\mathcal{K}^{\vee}[2]}\mathcal{E})^{\vee}$

Suppose that $X_{1}$ and $X_{2}$ are equivalent through an isomorphism

$X_{2}=M_{v}(X_{1}),$

i.e., the moduli space of simple sheaves $\mathcal{E}_{p},p\in X_{2}$, on
$X_{1}$ with Mukai vector

$v(\mathcal{E}_{p})=(r,D,s)\in\tilde{\mathrm{H}}(X,\mathbb{Z}).$

Here $r$ is the rank of $\mathcal{E}_{p}$, $D=c_{1}(\mathcal{E}_{p})$, and
$s=\chi(\mathcal{E}_{p})-r$. We assume there exists another Hodge class
$v^{\prime}\in\tilde{\mathrm{H}}(X_{1},\mathbb{Z})$ such that
$\left<v,v^{\prime}\right>=1$; in particular, $v$ is primitive. Let
$\mathcal{E}\rightarrow X_{1}\times X_{2}$ denote a universal sheaf; by
simplicity of the sheaves, $\mathcal{E}$ is unique up to tensoring by a line
bundle from $X_{2}$. We may use $\mathcal{E}$ as a kernel inducing a derived
equivalence between $X_{1}$ and $X_{2}$ [Huy06, 10.25]. Our formulas for
inverses are compatible with tensoring the kernel by line bundles from one of
the factors.

In searching for Fourier-Mukai kernels, cohomological Fourier-Mukai transforms
play a crucial role. Let $\omega_{i}\in\mathrm{H}^{4}(X_{i},\mathbb{Z})$
denote the point class and set [Muk87, §1], [Huy06, p. 128]

$Z_{\mathcal{K}}:=\pi_{1}^{*}(1+\omega_{1})\operatorname{ch}(\mathcal{K})\pi_{2}^{*}(1+\omega_{2})\in\mathrm{H}^{*}(X_{1}\times
X_{2},\mathbb{Z}),$

where the middle term is the Chern character. Then $Z_{\mathcal{K}}$ induces
an integral isomorphism of Hodge structures

$\phi_{\mathcal{K}}:\tilde{\mathrm{H}}(X_{1},\mathbb{Z})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\tilde{\mathrm{H}}(X_{2},\mathbb{Z})$

compatible with Mukai pairings; this is called the cohomological Fourier-Mukai
transform. For $\mathcal{E}\in D^{b}(X_{1})$, we have the identity

$\phi_{\mathcal{K}}(v(\mathcal{E}))=v(\Phi_{\mathcal{K}}(\mathcal{E})).$

We use $\psi_{\mathcal{K}}$ to denote the cohomological transform of
$\Psi_{\mathcal{K}}$.

Most cohomological Fourier-Mukai transforms are induced by kernels

######  Proposition 7.

[Orl97, HMS09] Given an orientation-preserving integral Hodge isometry

$\phi:\tilde{\mathrm{H}}(X_{1},\mathbb{Z})\rightarrow\tilde{\mathrm{H}}(X_{2},\mathbb{Z})$

there exists a derived equivalence

$\Phi_{\mathcal{K}}:D^{b}(X_{1})\rightarrow D^{b}(X_{2})$

such that $\phi$ is the cohomological Fourier-Mukai transform of
$\Phi_{\mathcal{K}}$.

Suppose that $(X_{1},f_{1})$ is a polarized K3 surfaces of degree $2r_{0}s$
where $r_{0}$ and $s$ are relatively prime positive integers. Let $d_{0}$ be
an integer prime to $r_{0}$ and fix the isotropic Mukai vector

$v_{0}=(r_{0},d_{0}f_{1},d_{0}^{2}s)\in\tilde{\mathrm{H}}(X_{1},\mathbb{Z}).$

Since $r_{0}$ and $d_{0}^{2}s$ are relatively prime, there exists a Mukai
vector $v^{\prime}=(m,0,n)$ such that $\left<v_{0},v^{\prime}\right>=1$. Let
$X_{2}=M_{v}(X_{1})$, also a K3 surface, and choose a universal sheaf
$\mathcal{E}\rightarrow X_{1}\times X_{2}$. Our goal is to describe the
induced isomorphism

$\phi_{\mathcal{E}}:\tilde{\mathrm{H}}(X_{1},\mathbb{Z})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\tilde{\mathrm{H}}(X_{2},\mathbb{Z}).$

Following [HL10, Ch. 8] and [Yos99, §2], the polarization on $X_{2}$ is given
by

$\det({\pi_{2}}_{*}(\mathcal{E}\otimes\mathcal{O}_{H}(s(r_{0}-2d_{0}))))^{\vee},\quad
H\in|f_{1}|,$

a primitive ample divisor $f_{2}$ on $X_{2}$. More generally, we have an
isomorphism of Hodge structures

$\mathrm{H}^{2}(X_{2},\mathbb{Z})=(v_{0}^{\vee})^{\perp}/\mathbb{Z}v_{0}^{\vee},$

where the perpendicular subspace is taken with respect to the Mukai pairing.

######  Proposition 8.

[Yos99] Choose integers $d_{1}$ and $\ell$ such that $sd_{0}d_{1}-r_{0}\ell=1$
and take $\mathcal{K}=\mathcal{E}\otimes\pi_{2}^{*}L$ for some line bundle $L$
on $X_{2}$. With respect to the bases

$(1,0,0),(0,f_{i},0),(0,0,1)\in\tilde{\mathrm{H}}(X_{i},\mathbb{Z})$

the matrix of the cohomological Fourier-Mukai transform takes the form

$\phi_{\mathcal{K}}:=\left(\begin{matrix}d_{0}^{2}s&2d_{0}sr_{0}&r_{0}\\\
d_{0}\ell&2d_{0}d_{1}s-1&d_{1}\\\ \ell^{2}r_{0}&2d_{1}s\ell
r_{0}&d_{1}^{2}s\end{matrix}\right).$

The inverse is obtained reversing the sign of the middle basis vector and
interchanging the role of $d_{0}$ and $d_{1}$:

$\left(\begin{matrix}d_{0}^{2}s&2d_{0}sr_{0}&r_{0}\\\
d_{0}\ell&2d_{0}d_{1}s-1&d_{1}\\\ \ell^{2}r_{0}&2d_{1}s\ell
r_{0}&d_{1}^{2}s\end{matrix}\right)\left(\begin{matrix}d_{1}^{2}s&-2d_{1}sr_{0}&r_{0}\\\
-d_{1}\ell&2d_{0}d_{1}s-1&-d_{0}\\\ \ell^{2}r_{0}&-2d_{0}s\ell
r_{0}&d_{0}^{2}s\end{matrix}\right)=I.$

The formula

$\phi_{\mathcal{K}}\psi_{\mathcal{K}^{\vee}}=I$

is the cohomological realization of the identity

$\Phi_{\mathcal{K}}\Psi_{\mathcal{K}^{\vee}[2]}=I.$

The third column of $\phi_{\mathcal{K}}^{-1}$ is the Mukai vector
$v_{0}^{\vee}$, as

$\Phi_{\mathcal{K}}^{-1}(\mathcal{O}_{p})=\mathcal{E}_{p}^{\vee},\quad
p=[\mathcal{E}_{p}]\in X_{2}=M_{v_{0}}(X_{1}).$

######  Example 9.

Suppose that $(X_{1},f_{1})$ is a degree $12$ K3 surface. Consider the
isotropic Mukai vector $v=(2,f_{1},3)$ so that

$X_{2}:=M_{v}(X_{1})$

is also a K3 surface derived equivalent to $X_{1}$. Taking

$r_{0}=2,s=3,d_{0}=1,d_{1}=\ell=1$

we obtain

$\displaystyle(1,0,0)$ $\displaystyle\mapsto(3,f_{2},2)$
$\displaystyle(0,f_{1},0)$ $\displaystyle\mapsto(12,5f_{2},12)$
$\displaystyle(0,0,1)$ $\displaystyle\mapsto(2,f_{2},3)$

with matrix

(4.1) $\varphi:=\left(\begin{matrix}3&12&2\\\ 1&5&1\\\
2&12&3\end{matrix}\right).$

The determinant is $1$ with one eigenvector $(1,0,-1)$ with eigenvalue $1$;
thus this is orientation preserving. Note that

$(2,-f_{1},3)\mapsto(0,0,1)$

whence

$X_{1}=M_{(2,f_{2},3)}(X_{2}),\quad X_{2}=M_{(2,-f_{1},3)}(X_{1}).$

The fact that $(1,0,-1)$ has eigenvalue $1$ gives

$X_{1}^{[2]}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X_{2}^{[2]}.$

##  5\. Generalities concerning involutions on K3 surfaces

Let $i:X\rightarrow X$ be an involution on a complex projective K3 surface,
which acts faithfully on $\mathrm{H}^{2}(X,\mathbb{Z})$ by the Torelli
Theorem. It is symplectic (resp. anti-symplectic) if

$i^{*}\omega=\omega\quad(\text{resp. }-\omega),$

where $\omega$ is a holomorphic two-form. Nikulin [Nik79a] showed that any
symplectic involution fixes eight isolated points and that all such
involutions are topologically conjugate; these are the Nikulin involutions
studied in Section 8. An involution without fixed points was classically known
to be an Enriques involution arising from a double cover $X\rightarrow S$ of
an Enriques surface.

The case of anti-symplectic involutions with fixed points is more complicated.
Nikulin enumerated 74 cases beyond the Enriques case; see [AN06, BH21, AE22,
Ale22] for details of the various cases.

Given an anti-symplectic involution $i:X\rightarrow X$ on a K3 surface, we
recall the Nikulin invariants $(r,a,\delta)$ [AE22, §2]: Let $r$ denote the
rank of the lattice

$S=\mathrm{H}^{2}(X,\mathbb{Z})^{i=1},$

which is indefinite if $r>1$. We are using the fact that transcendental
classes of $X$ are anti-invariant under $i$, as the quotient $X/i$ admits no
holomorphic two-form. We write

$T=\mathrm{H}^{2}(X,\mathbb{Z})^{i=-1}=S^{\perp}$

for the complementary lattice with signature $(2,20-r)$, which is indefinite
if $r<20$. The discriminant group $d(S)\simeq d(T)$ is a $2$-elementary group;
its rank is denoted by $a$. This group comes with a quadratic form

$q_{S}:d(S)\rightarrow\mathbb{Q}/2\mathbb{Z}.$

The coparity $\delta$ equals $0$ if $q_{S}(x)\in\mathbb{Z}$ for each $x\in
d(S)$ and equals $1$ otherwise.

We relate this to geometric invariants. For an anti-symplectic involution,
there are no isolated fixed points so the fixed locus $R=X^{i}$ is of pure
dimension one or empty. Suppose there are $k+1$ irreducible components, with
genera summing to $g$. Then we have cf. [AE22, p.5]

$g=11-(r+a)/2\quad k=(r-a)/2,$

excluding the Enriques case $(r,k,\delta)=(10,10,0)$.

Nikulin classifies even indefinite $2$-elementary lattices $\mathrm{L}$. They
are determined uniquely by $(r,a,\delta)$ and
$\mathrm{O}(\mathrm{L})\rightarrow\operatorname{Aut}(d(\mathrm{L}))$ is
surjective. In the definite case, a priori there are multiple classes in each
genus but this is not relevant for our applications. Indeed, the possibilities
include

  * •

$r=a=1$: $X$ is a double cover of $\mathbb{P}^{2}$ branched along a sextic
plane curve.

  * •

The case where $T$ is definite $(r=20,a=2,g=0,k=9)$, we have
$d(T)=\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$ thus is equal to

$\left(\begin{matrix}2&0\\\ 0&2\end{matrix}\right).$

Even in this case, automorphisms of the discriminant group are realized by
automorphisms of the lattice.

######  Theorem 10 (Alexeev-Nikulin).

For each admissible set of invariants $(r,a,\delta)$, there is a unique
orthogonal pair of lattices (S,T) embedded in the K3 lattice $\Lambda$, up to
automorphisms of $\Lambda$. There are 75 such cases.

######  Corollary 11.

Any equivariant derived equivalence of K3 surfaces with anti-symplectic
involutions induces an equivariant isomorphism between the underlying K3
surfaces.

###### Proof.

Suppose that $(X_{1},i_{1})$ and $(X_{2},i_{2})$ are derived equivalent,
compatibly with their anti-symplectic involutions.

Indeed, derived equivalence shows that the invariant (resp. anti-invariant)
sublattices of the Picard group are stably equivalent (resp. equivalent):

$\operatorname{Pic}(X_{1})^{i_{1}=1}\oplus
U\simeq\operatorname{Pic}(X_{2})^{i_{2}=1}\oplus
U,\quad\operatorname{Pic}(X_{1})^{i_{1}=-1}\simeq\operatorname{Pic}(X_{2})^{i_{2}=-1}.$

Since the possibilities for the invariant sublattices are characterized by
their $2$-adic invariants, we have

$\operatorname{Pic}(X_{1})^{i_{1}=1}\simeq\operatorname{Pic}(X_{2})^{i_{2}=1}.$

We have already observed that all the possible isomorphisms between their
discriminants

$\left(d(\operatorname{Pic}(X_{1})^{i_{1}=1}),q_{1}\right)\simeq\left(d(\operatorname{Pic}(X_{2})^{i_{2}=1}),q_{2}\right)$

are realized by isomorphisms of the lattices. In particular, there exists a
choice compatible with the isomorphism

$\mathrm{H}^{2}(X_{1},\mathbb{Z})^{i_{1}=-1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathrm{H}^{2}(X_{2},\mathbb{Z})^{i_{2}=-1}$

induced by the derived equivalence. Thus we obtain isomorphisms on middle
cohomology, compatible with the involutions. The Torelli Theorem gives an
isomorphism $X_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}X_{2}$
respecting the involutions. ∎

######  Corollary 12.

Let $(X_{1},\sigma_{1})$ and $(X_{2},\sigma_{2})$ denote K3 surfaces with
involutions that are $C_{2}$-equivariantly derived equivalent. If
$X_{1}/\sigma_{1}$ is rational then $X_{2}/\sigma_{2}$ is rational as well.

Indeed, the rationality of the quotient forces the involution to be anti-
symplectic.

######  Example 13.

Having an anti-symmetric involution is not generally a derived property. For
example, consider Picard lattices

$A_{1}=\left(\begin{matrix}2&13\\\ 13&12\end{matrix}\right)\quad
A_{2}=\left(\begin{matrix}8&15\\\ 15&10\end{matrix}\right).$

These forms are stably equivalent but not isomorphic. As in Remark 5 – see
[HT17, Sec. 2.3] for details – choose derived equivalent K3 surfaces $X_{1}$
and $X_{2}$ with $\operatorname{Pic}(X_{1})=A_{1}$ and
$\operatorname{Pic}(X_{2})=A_{2}$. Note that $A_{2}$ does not represent two
and admits no involution acting via $\pm 1$ on $d(A_{2})$; thus $X_{2}$ does
not admit an involution.

This should be compared with Proposition 28: Having an Enriques involution is
a derived invariant.

##  6\. Orientation reversing conjugation

We continue to assume that $i$ is an anti-symplectic involution on a K3
surfaces $X$. As we have seen,

$T(X)\subset\mathrm{H}^{2}(X,\mathbb{Z})^{i=-1}$

with complement $\operatorname{Pic}(X)^{i=-1}$, which is negative definite by
the Hodge index theorem.

Recall that Orlov’s Theorem [Orl97, §3] asserts that for K3 surfaces (without
group action) isomorphisms of transcendental cohomology lift to derived
equivalences. Given K3 surfaces $(X_{1},i_{1})$ and $(X_{2},i_{2})$ with anti-
symplectic involutions of the same type in the sense of Alexeev-Nikulin, the
existence of an isomorphism

$T(X_{1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T(X_{2})$

seldom induces an equivariant derived equivalence; a notable exception is the
case where the anti-invariant Picard group has rank zero or one. We only have
that

$\operatorname{Pic}(X_{1})^{i_{1}=-1},\quad\operatorname{Pic}(X_{2})^{\epsilon_{2}=-1}$

are stably equivalent – compatibly with the isomorphism on the discriminant
groups of the transcendental lattices – but not necessarily isomorphic.

In light of this, we propose an orientation reversing conjugation of actions,
with a view toward realizing isomorphisms of transcendental cohomology.

Assume that $\operatorname{Pic}(X_{1})^{i_{1}=-1}$ and
$\operatorname{Pic}(X_{2})^{i_{2}=-1}$ are not isomorphic, so there is no
$C_{2}$-equivariant derived equivalence

$D^{b}(X_{1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}D^{b}(X_{2})$

taking $i_{1}$ to $i_{2}$, by Corollary 11. However, let

$\operatorname{dual}_{k}:D^{b}(X_{k})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}D^{b}(X_{k}),\quad
k=1,2,$

denote the involution

$\mathcal{E}_{*}\mapsto\mathcal{E}^{\vee}_{*}.$

Note that shift and duality commute with each other and with any automorphism
of the K3 surface. The action of $\operatorname{dual}_{k}$ on the Mukai
lattice $\tilde{\mathrm{H}}(X_{k},\mathbb{Z})$ is trivial in degrees $0$ and
$4$ and multiplication by $-1$ in degree two. Recall that shift acts via $-1$
in all degrees, so composition with $\operatorname{dual}_{k}$ is trivial in
degree $2$ and multiplication by $-1$ in degrees $0$ and $4$.

We propose a general definition and then explain how it is related to our
analysis of quadratic forms with involution:

######  Definition 14.

Let $(X_{1},i_{1})$ and $(X_{2},i_{2})$ be smooth projective varieties with
involution, of dimension $n$ with trivial canonical class. They are skew
equivalent if there is a kernel $\mathcal{K}$ on $X_{1}\times X_{2}$, inducing
an equivalence between $X_{1}$ and $X_{2}$, such that

(6.1) $(i_{1}^{*},i_{2}^{*})\mathcal{K}=\mathcal{K}^{\vee}[n].$

Note that this dualization coincides with the relative dualizing complex for
both projections $\pi_{1}$ and $\pi_{2}$.

Suppose again that $X_{1}$ and $X_{2}$ are K3 surfaces and
$\mathcal{K}=\mathcal{E}[1]$ for a universal sheaf

$\mathcal{E}\rightarrow X_{1}\times X_{2}$

associated with an isomorphism $X_{2}=M_{v}(X_{1})$. Then relation (6.1) (with
$n=2$) translates into

(6.2) $i_{1}^{*}\mathcal{E}_{i_{2}(x_{2})}\simeq(\mathcal{E}_{x_{2}})^{\vee}.$

######  Proposition 15.

Let $(X_{1},i_{1})$ and $(X_{2},i_{2})$ be K3 surfaces with involutions. Then
the following are equivalent

  * •

$(X_{1},i_{1})$ and $(X_{2},i_{2})$ are skew derived equivalent;

  * •

there exists an orientation-preserving equivalence of Mukai lattices

$\phi:\tilde{\mathrm{H}}(X_{1},\mathbb{Z})\longrightarrow\tilde{\mathrm{H}}(X_{2},\mathbb{Z}),$

satisfying

(6.3) $\phi(i_{1}^{*}(v^{\vee}))=(i_{2}^{*}\phi(v))^{\vee}.$

As duality and pull back commute with each other, the order of these
operations in (6.3) is immaterial. Furthermore, if $\phi$ satisfies this
relation then so does $-\phi$.

###### Proof.

The forward implication is clear. Indeed, the cohomological Fourier-Mukai
transform $\phi_{\mathcal{K}}$ of a skew equivalence satisfies

$(i_{1},i_{2})^{*}\phi_{\mathcal{K}}=\phi_{\mathcal{K}^{\vee}}$

but $\phi_{\mathcal{K}^{\vee}}$ differs from $\phi_{\mathcal{K}}$ by the
involution acting via $+1$ on $\mathrm{H}^{0}$ and $\mathrm{H}^{4}$ and $-1$
on $\mathrm{H}^{2}$. Thus

$\phi_{\mathcal{K}}:\tilde{\mathrm{H}}(X_{1},\mathbb{Z})\longrightarrow\tilde{\mathrm{H}}(X_{2},\mathbb{Z})$

is an isomorphism equivariant under the prescribed “skew” involutions.

For the reverse implication, we consider the cohomological Fourier-Mukai
transform

$\phi:\tilde{\mathrm{H}}(X_{1},\mathbb{Z})\longrightarrow\tilde{\mathrm{H}}(X_{2},\mathbb{Z}).$

Proposition 7 yields a kernel $\mathcal{K}$ such that
$\phi=\phi_{\mathcal{K}}$. We make this more explicit.

Set

$v=(\phi^{-1}(0,0,1))^{\vee}$

with view toward relating $\mathcal{K}$ to a universal bundle

$\mathcal{E}\rightarrow X_{1}\times X_{2},$

where $X_{2}$ is a moduli space of bundles on $X_{1}$. Write $v=(r,f_{1},s)$;
if $r<0$, replace $\phi$ by $-\phi$. This leaves (6.3) unchanged and
corresponds to replacing $\Phi_{\mathcal{K}}$ by $\Phi_{\mathcal{K}}$ pre-
composed (or post-composed) by a shift on $X_{1}$ (or $X_{2}$).

Thus we may assume $v_{0}=(r_{0},f_{1},s)$ with $r_{0}>0$ and consider a
universal bundle

$\mathcal{E}\rightarrow X_{1}\times M_{v_{0}}(X_{1})\simeq X_{1}\times X_{2}.$

We therefore have (see Proposition 8 for the formula and our basis
conventions)

$\phi_{\mathcal{E}}=\left(\begin{matrix}d_{0}^{2}s&2d_{0}sr_{0}&r_{0}\\\
d_{0}\ell&2d_{0}d_{1}s-1&d_{1}\\\ \ell^{2}r_{0}&2d_{1}s\ell
r_{0}&d_{1}^{2}s\end{matrix}\right)$

and

$\phi_{\mathcal{K}}=\left(\begin{matrix}d_{0}^{2}s&2d_{0}sr_{0}&r_{0}\\\
d_{0}\hat{\ell}&2d_{0}\hat{d_{1}}s-1&\hat{d_{1}}\\\
\hat{\ell}^{2}r_{0}&2\hat{d_{1}}s\hat{\ell}r_{0}&\hat{d}_{1}^{2}s\end{matrix}\right).$

Here we have

$sd_{0}d_{1}-r_{0}\ell=sd_{0}\hat{d_{1}}-r_{0}\hat{\ell}=1,$

whence

$\hat{\ell}=\ell+Nsd_{0}\quad\hat{d_{1}}=d_{1}+Nr_{0}.$

Consider the autoequivalence on $X_{2}$ obtained by tensoring with the
invertible sheaf $\mathcal{O}_{X_{2}}(Nf_{2})$. This has cohomological
Fourier-Mukai transform with matrix

$t^{N}:=\left(\begin{matrix}1&0&0\\\ N&1&0\\\
N^{2}r_{0}s&2Nr_{0}s&1\end{matrix}\right).$

Note however that

$t^{N}\phi_{\mathcal{E}}=\phi_{\mathcal{K}}$

and we can renormalize $\mathcal{E}$ so that it has cohomological Fourier-
Mukai transform $\phi$. Specifically, there exists an isomorphism

$X_{2}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}M_{v_{0}}(X_{1})$

such that the pullback of the universal sheaf $\mathcal{E}$ to $X_{1}\times
X_{2}$ induces $\phi$. We analyze how $i_{1}$ and $i_{2}$ act on
$\mathcal{E}$, keeping in mind the functional relation. We have that

$\mathcal{E},\quad(i_{1},i_{2})^{*}\mathcal{E}^{\vee}$

are both universal bundles on $X_{1}\times X_{2}$ with the same numerical
invariants. The uniqueness of such bundles gives an isomorphism

$\xi:\mathcal{E}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}(i_{1},i_{2})^{*}\mathcal{E}^{\vee}$

over $X_{1}\times X_{2}$, unique up to a scalar. Note there are two
distinguished normalizations of this scalar, for which the composition

$(i_{1},i_{2})^{*}\xi^{\vee}\circ\xi:\mathcal{E}\rightarrow\mathcal{E}$

is the identity. For purposes of establishing derived equivalences, the choice
of normalization is immaterial. ∎

######  Corollary 16.

Under the assumptions above, the functors $\operatorname{dual}_{1}\circ i_{1}$
and $\operatorname{dual}_{2}\circ i_{2}$ are $C_{2}$-equivariantly derived
equivalent.

######  Remark 17.

As we recalled in Section 3, derived equivalences respect orientations on the
Mukai lattice [HMS09]. The orientation reversing conjugation violates the
orientation condition, in a prescribed way. Duality is the archetypal
orientation-reversing Hodge isogeny.

In Sections 7 and 10 we give examples of such equivalences.

##  7\. Rational quotients and skew equivalence

Our first task is to give examples of skew equivalences. Proposition 15 and
the discussion preceding it reduce this to exhibiting lattice-polarized K3
surfaces with involution $(X_{1},i_{1})$ and $(X_{2},i_{2})$, such that the
anti-invariant Picard groups are stably equivalent but inequivalent.

Specifically, we assume $X_{1}$ and $X_{2}$ are degree two K3 surfaces with

$\operatorname{Pic}(X_{j})=\mathbb{Z}h_{j}\oplus A_{j}(-1),\quad h_{j}^{2}=2,$

where the involutions fix the $h_{j}$ and reverse signs on $A_{j}$’s. If
$A_{1}$ and $A_{2}$ are stably-equivalent, inequivalent positive definite
lattices then $(X_{1},i_{1})$ and $(X_{2},i_{2})$ are skew equivalent.

In contrast to ordinary equivalences (see 11) we do have antisymplectic
actions with nontrivial skew equivalences. The resulting quotients are
rational surfaces, indeed, $\mathbb{P}^{2}$.

######  Example 18 (Explicit matrices).

The matrices, in the basis $p_{j},q_{j}$, for $j=1,2$, are given by

$A_{1}:=\begin{pmatrix}4&1\\\ 1&12\end{pmatrix},\quad
A_{2}:=\begin{pmatrix}6&1\\\ 1&8\end{pmatrix}.$

We extract a stable isomorphism

$A_{1}\oplus\mathrm{U}\simeq
A_{2}\oplus\mathrm{U},\quad\mathrm{U}=\left<u_{1},v_{1}\right>,\text{ with
matrix }\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}.$

First, we give an isomorphism

$A_{1}\oplus\left<e_{1}\right>\simeq A_{2}\oplus\left<e_{2}\right>,\quad
e_{1}^{2}=-2.$

We put

$p_{1}\mapsto p_{2}+e_{2},$

and claim that the orthogonal complements to these are equivalent indefinite
lattices. Indeed,

$p_{1}^{\perp}=\left<p_{1}-4q_{1},e_{1}\right>=\begin{pmatrix}188&0\\\
0&-2\end{pmatrix},$ $\displaystyle(p_{2}+e_{2})^{\perp}$
$\displaystyle=\left<p_{2}-6q_{2},2q_{2}+e_{2}\right>=\begin{pmatrix}282&-94\\\
-94&30\end{pmatrix}$
$\displaystyle=\left<p_{2}+3e_{2},2q_{2}+e_{2}\right>=\begin{pmatrix}-12&-4\\\
-4&30\end{pmatrix}$

These are equivalent via Gaussian cycles of reduced forms

$\begin{array}[]{ccccccccc}&0&&18&&8&&4&\\\ 188&&-2&&26&&-12&&30\end{array}$

where the indicated basis elements are

$p_{1}-4q_{1},\quad e_{1},\quad p_{1}-4q_{1}-9e_{1},\quad
p_{1}-4q_{1}-10e_{1},\quad 2(p_{1}-4q_{1})-19e_{1}.$

The composed isomorphism is

$\displaystyle p_{1}-4q_{1}-10e_{1}$ $\displaystyle\mapsto p_{2}+3e_{2},$
$\displaystyle 2(p_{1}-4q_{1})-19e_{1}$ $\displaystyle\mapsto 2q_{2}+e_{2}$
$\displaystyle p_{1}$ $\displaystyle\mapsto p_{2}+e_{2}$ $\displaystyle e_{1}$
$\displaystyle\mapsto(2q_{2}+e_{2})-2(p_{2}+3e_{2})=2(q_{2}-p_{2})-5e_{2}$
$\displaystyle q_{1}$ $\displaystyle\mapsto 5(p_{2}-q_{2})+12e_{2}.$

We extend the isomorphism above where $e_{i}=u_{i}-v_{i}$

$\displaystyle u_{1}+v_{1}$ $\displaystyle\mapsto u_{2}+v_{2}$ $\displaystyle
u_{1}-v_{1}$ $\displaystyle\mapsto 2(q_{2}-p_{2})-5(u_{2}-v_{2})$
$\displaystyle p_{1}$ $\displaystyle\mapsto p_{2}+(u_{2}-v_{2})$
$\displaystyle q_{1}$ $\displaystyle\mapsto 5(p_{2}-q_{2})+12(u_{2}-v_{2})$

whence we have

$\displaystyle u_{1}$ $\displaystyle\mapsto(q_{2}-p_{2})-2u_{2}+3v_{2}$
$\displaystyle v_{1}$ $\displaystyle\mapsto(p_{2}-q_{2})+3u_{2}-2v_{2}.$

##  8\. Nikulin involutions

### General properties

An involution $\iota$ on a K3 surface $X$ over $\mathbb{C}$ preserving the
symplectic form is called a Nikulin involution. We recall basic facts
concerning such involutions, following [vGS07]:

  * •

$\iota$ has 8 isolated fixed points;

  * •

the (resolution of singularities) $Y\rightarrow X/\iota$ is a K3 surface
fitting into a diagram

$\begin{array}[]{ccl}X&\stackrel{{\scriptstyle\beta}}{{\leftarrow}}&\widetilde{X}\\\
\downarrow&&\downarrow\scriptsize{\pi}\\\ X/\iota&\leftarrow&Y\end{array}$

where $\beta$ blows up the fixed points and the vertical arrows have degree
two;

  * •

the action of $\iota$ on $\mathrm{H}^{2}(X,\mathbb{Z})$ is uniquely
determined, and there is a decomposition

$\mathrm{H}^{2}(X,\mathbb{Z})=(\mathrm{U}^{\oplus
3})_{1}\oplus(\mathrm{E}_{8}(-1)\oplus\mathrm{E}_{8}(-1))_{P},$

where the first term is invariant and the second is a permutation module for
$\iota$;

  * •

the invariant and the anti-invariant parts of $\mathrm{H}^{2}$ take the form:

$\mathrm{H}^{2}(X,\mathbb{Z})^{\iota=1}\simeq\mathrm{U}^{3}\oplus\mathrm{E}_{8}(-2),\quad\mathrm{H}^{2}(X,\mathbb{Z})^{\iota=-1}=\mathrm{E}_{8}(-2)$

Let $E_{1},\ldots,E_{8}$ denote the exceptional divisors of $\beta$ and
$N_{1},\ldots,N_{8}$ the corresponding $(-2)$-curves on $Y$. The union $\cup
N_{i}$ is the branch locus of $\pi$ so there is a divisor

$\hat{N}=(N_{1}+\cdots+N_{8})/2$

saturating $\left<N_{1},\ldots,N_{8}\right>\subset\operatorname{Pic}(Y)$; the
minimal primitive sublattice containing these divisors is called the Nikulin
lattice, and is denoted by $\mathrm{N}$. We have [vGS07, Prop. 1.8]

$\begin{array}[]{rcl}\pi_{*}:\mathrm{H}^{2}(\widetilde{X},\mathbb{Z})&\rightarrow&\mathrm{H}^{2}(Y,\mathbb{Z})\\\
\mathrm{U}^{3}\oplus\mathrm{E}_{8}(-1)\oplus\mathrm{E}_{8}(-1)\oplus\left<-1\right>^{8}&\rightarrow&\mathrm{U}(2)^{3}\oplus\mathrm{N}\oplus\mathrm{E}_{8}(-1)\\\
(u,x,y,z)&\mapsto&(u,z,x+y)\end{array}$

and

$\begin{array}[]{rcl}\pi^{*}:\mathrm{H}^{2}(Y,\mathbb{Z})&\rightarrow&\mathrm{H}^{2}(\widetilde{X},\mathbb{Z})\\\
\mathrm{U}(2)^{3}\oplus\mathrm{N}\oplus\mathrm{E}_{8}(-1)&\rightarrow&\mathrm{U}^{3}\oplus\mathrm{E}_{8}(-1)\oplus\mathrm{E}_{8}(-1)\oplus\left<-1\right>^{8}\\\
(u,n,x)&\mapsto&(2u,x,x,2\tilde{n})\end{array}$

where if $n=\sum n_{i}N_{i}$ then $\tilde{n}=\sum n_{i}E_{i}$. Thus we obtain
a distinguished saturated sublattice

$\mathrm{E}_{8}(-2)\subset\operatorname{Pic}(X)$

that coincides with the $\iota=-1$ piece.

######  Proposition 19.

Fix a lattice $\mathrm{L}$ containing $\mathrm{E}_{8}(-2)$ as a primitive
sublattice; assume $\mathrm{L}$ arises as the Picard lattice of a projective
K3 surface. Then there exists a K3 surface $X$ with Nikulin involution $\iota$
such that

$\mathrm{L}=\operatorname{Pic}(X)\supset\operatorname{Pic}(X)^{\iota=-1}=\mathrm{E}_{8}(-2).$

###### Proof.

Let $\mathrm{A}$ denote the orthogonal complement of $\mathrm{E}_{8}(-2)$ is
$\mathrm{L}$. There is a unique involution $\iota$ on $\mathrm{L}$ with

$\mathrm{L}^{\iota=1}=\mathrm{A},\quad\mathrm{L}^{\iota=-1}=\mathrm{E}_{8}(-2).$

Now $\iota$ acts trivially on $d(\mathrm{L})$ – keep in mind
$d(\mathrm{E}_{8}(-2))$ is a two-elementary group – so we may naturally extend
$\iota$ to the full K3 lattice. (It acts trivially on $\mathrm{L}^{\perp}.$)
These lattice-polarized K3 surfaces form our family

Nikulin [Nik79a, §4] explains how to get involutions for generic K3 surfaces
with lattice polarization $\mathrm{L}$. Choose such a surface $X$ such that
$\operatorname{Pic}(X)=\mathrm{L}$ – a very general member of the family has
this property. Clearly $X$ is projective – it admits divisors with positive
self-intersection. We claim there is an ample divisor $H\in\mathrm{A}$.
Indeed, the ample cone of $X$ is characterized as the chamber of the cone of
positive divisors by the group generated by reflections associated with
indecomposable $(-2)$-classes $E$ of positive degree [LP81]. Each $(-2)$-class
$E$ is perpendicular to a unique ray in

$\mathrm{A}\otimes\mathbb{R}\cap\\{\text{ cone of positive divisors }\\}$

generated by an element $a_{E}\in\mathrm{A}$. Note that $\mathrm{A}$ cannot be
contained in $E^{\perp}$ as $\mathrm{E}_{8}(-2)$ has no $(-2)$-classes. We
conclude that $\mathrm{A}$ meets each chamber in the decomposition of the
positive cone – it cannot be separated from the ample cone by any of the
$\mathrm{E}^{\perp}$.

Once we have the ample cone, we can extract the automorphism group of $X$ via
the Torelli Theorem: It consists of the Hodge isometries taking the ample cone
to itself. In particular, any Hodge isometry fixing $H$ is an automorphism.
Thus $\iota$ is an automorphism of $X$. ∎

######  Proposition 20.

Let $\mathrm{L}$ be an even hyperbolic lattice containing $\mathrm{E}_{8}(-2)$
as a saturated sublattice. Assume that $d(\mathrm{L})$ has rank at most $11$.
Then $\mathrm{L}$ is unique in its genus and the homomorphism

${\mathrm{O}}(\mathrm{L})\rightarrow{\mathrm{O}}(q_{\mathrm{L}})$

is surjective.

The condition on the rank of $d(L)$ is satisfied for Picard lattices of K3
surfaces $X$. We have

$\operatorname{Pic}(X)\subset\mathrm{U}^{\oplus
3}\oplus\mathrm{E}_{8}(-1)^{\oplus 2}$

which has rank $22$; $d(\operatorname{Pic}(X))\simeq d(T(X))$ so both groups
are generated by $\leq 11$ elements.

###### Proof.

We apply Proposition 2. For odd primes $p$, the conditions are easily checked
as the rank $r$ of $\mathrm{L}$ exceeds the rank of the $p$-primary part
$d(\mathrm{L})$. If $r\geq 12$ then the discriminant group is generated by
$\leq 10$ elements and we are done. Thus we focus on the $p=2$ case with
$r=9,10$, or $11$.

Let $\mathrm{A}$ denote the orthogonal complement to $\mathrm{E}_{8}(-2)$ in
$\mathrm{L}$. The overlattice

$\mathrm{L}\supset\mathrm{A}\oplus\mathrm{E}_{8}(-2)$

corresponds to an isotropic subgroup

$H\subset d(\mathrm{A})\oplus d(\mathrm{L})$

with respect to $q_{\mathrm{A}}\oplus q_{\mathrm{L}}$. Projection maps $H$
injectively into each summand – we may interpret these projections as kernels
of the natural maps

$d(\mathrm{A})\rightarrow d(\mathrm{L}),\quad d(\mathrm{E}_{8}(-2))\rightarrow
d(\mathrm{L}).$

Thus $H$ is a $2$-elementary group, of rank at most three. It follows that
$d(\mathrm{L})$ contains at least five copies of $\mathbb{Z}/2\mathbb{Z}$.
Remark 3 shows this validates the hypothesis of Proposition 2. ∎

The assumption on the rank of the discriminant groups can be replaced by
bounds on its order [CS99, Cor. 22, p. 395] – at least for purposes of showing
there is one class in each genus.

### Rank nine examples

We focus on examples with Picard rank nine, following [vGS07, Prop. 2.2] which
lists the possible lattices. Suppose that
$\operatorname{Pic}(X)^{\iota=1}=\mathbb{Z}f$ with $f^{2}=2d$, which is
necessarily ample as there are no $(-2)$-classes in

$\operatorname{Pic}(X)^{\iota=-1}=\mathrm{E}_{8}(-2).$

We have the lattice

$\Lambda:=\left(2d\right)\oplus\mathrm{E}_{8}(-2),$

for all $d$. For even $d$ we have the index-two overlattice
$\widetilde{\Lambda}\supset\Lambda$, generated by

$\frac{f+e}{2},$

where $f$ is a generator of $\left(2d\right)$ and $e\in\mathrm{E}_{8}(-2)$ is
a primitive element with

$\left(e,e\right)=\begin{cases}-4&\text{ if }d=4m+2\\\ -8&\text{ if
}d=4m.\end{cases}$

We are using the fact that the lattice $\mathrm{E}_{8}$ has primitive vectors
of lengths $2$ and $4$. Using the shorthand

$q(v)=q_{\mathrm{E}_{8}(-2)}(v)\pmod{2\mathbb{Z}}$

elements $0\neq v\in e_{8}(-2):=d(\mathrm{E}_{8}(-2))$ are of two types

  * •

$120$ elements $v$ with $q(v)=1$ ($A_{1}+E_{7}$ type);

  * •

$135$ elements $v$ with $q(v)=0$ ($D_{8}$ type).

Note that $\widetilde{\Lambda}$ is the unique such overlattice such that
$\mathrm{E}_{8}(-2)$ remains saturated.

######  Proposition 21.

Let $(X_{1},f_{1})$ and $(X_{2},f_{2})$ be polarized K3 surfaces of degree
$2d$, derived equivalent via specialization of the construction in Remark 5.
If $X_{1}$ admits a Nikulin involution fixing $f_{1}$ then

  * •

$X_{2}$ admits a Nikulin involution fixing $f_{2}$;

  * •

there is an isomorphism

$\varphi:X_{1}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}X_{2}.$

###### Proof.

The derived equivalence induces an isomorphism of lattices with Hodge
structure

$\mathrm{H}^{2}(X_{1},\mathbb{Z})\supset f_{1}^{\perp}\simeq
f_{2}^{\perp}\subset\mathrm{H}^{2}(X_{2},\mathbb{Z}),$

which means that $f_{2}^{\perp}\cap\operatorname{Pic}(X_{2})$ contains a
sublattice isomorphic to $E_{8}(-2)$. Thus there exists a Hodge involution

$\iota_{2}^{*}:\mathrm{H}^{2}(X_{2},\mathbb{Z})\rightarrow\mathrm{H}^{2}(X_{2},\mathbb{Z})$

with anti-invariant summand equal to this copy of $\mathrm{E}_{8}(-2)$. The
Torelli Theorem – see [vGS07, Prop. 2.3] – shows that $X_{2}$ admits an
involution $\iota_{2}:X_{2}\rightarrow X_{2}$.

Isomorphisms of K3 surfaces specialize in families [MM64, ch. I]. This reduces
us to proving the result when the $X_{k}$ have Picard rank nine, putting us in
the case of Proposition 20. The Counting Formula of [HLOY04, §2] – using the
conclusions of Proposition 20 – implies that all Fourier-Mukai partners of
$X_{1}$ are isomorphic to $X_{1}$. ∎

######  Remark 22.

We are not asserting that $\varphi^{*}f_{2}=f_{1}$! Suppose that $X_{1}$ and
$X_{2}$ have Picard rank nine, the minimal possible rank. Then

$\varphi^{*}f_{2}\equiv\alpha f_{1}\pmod{\mathrm{E}_{8}(-2)}$

where $\alpha\pmod{4d}$ is the corresponding solution to congruence (3.1).

Thus we obtain nontrivial derived equivalence among Nikulin surfaces even in
rank nine!

### Rank ten examples

Turning to rank ten, we offer a generalization of [vGS07, Prop. 2.3]:

######  Proposition 23.

Fix a rank two indefinite even lattice $\mathrm{A}$ and an even extension

$\mathrm{L}\supset\mathrm{A}\oplus\mathrm{E}_{8}(-2)$

invariant under $\iota$; here $\iota$ fixes $\mathrm{A}$ and acts by
multiplication by $-1$ on $\mathrm{E}_{8}(-2)$. Then there exists a K3 surface
$X$ with Nikulin involution $\iota$ such that

$\mathrm{A}=\operatorname{Pic}(X)^{\iota=1}\subset\operatorname{Pic}(X)=\mathrm{L}\supset\operatorname{Pic}(X)^{\iota=-1}=\mathrm{E}_{8}(-2).$

###### Proof.

The lattice $\mathrm{L}$ embeds uniquely into the K3 lattice by Proposition 4.
Proposition 19 gives the desired K3 surface with involution. ∎

We observed in Proposition 20 that the lattice $\mathrm{L}$ are unique in
their genus and admit automorphisms realizing the full group
${\mathrm{O}}(d(\mathrm{L}))$. Repeating the reasoning for Proposition 21 we
find:

######  Proposition 24.

A K3 surface $X$ with involution $\iota_{1}$, produced following Proposition
23 applied to $\mathrm{A}_{1}$, will have a second involution $\iota_{2}$
associated with $\mathrm{A}_{2}$. Moreover $(X,\iota_{1})$ and $(X,\iota_{2})$
are not equivariantly derived equivalent.

We elaborate on the overlattices $\mathrm{L}$ arising in the assumptions of
Proposition 23. What lattices may arise from a given $\mathrm{A}$? Each
$\mathrm{L}$ arises from a $2$-elementary

$H\subset d(\mathrm{A})\oplus e_{8}(-2)$

isotropic with respect to $q_{\mathrm{A}}\oplus q_{\mathrm{E}_{8}(-2)}$.

We consider the orbits of

$H\simeq(\mathbb{Z}/2\mathbb{Z})^{2}\subset e_{8}(-2)$

under automorphisms of the lattice. These reflect possible quadratic forms on
$(\mathbb{Z}/2\mathbb{Z})^{2}$. We enumerate the possibilities, relying on
description of maximal subgroups of the simple group of
${\mathrm{O}}^{+}_{8}(2)$ (automorphisms of $e_{8}(-2)$) [CCN+85, p. 85] and
subgroups of $W(\mathrm{E}_{8})$ (a closely related group) associated with
reflections [DPR13, Table 5]. For the reader’s reference, we list the root
systems associated with the subgroups in parentheses:

  1. (1)

isotropic subspaces, where $q|H$ is trivial – $1575$ elements
($\mathrm{D}_{4}+\mathrm{D}_{4}$ type);

  2. (2)

rank one subspaces, where $q|H$ has a kernel, e.g., $q(x,y)=x^{2}$ –
$3780=28\times 135$ elements ($\mathrm{A}_{1}+\mathrm{A}_{1}+\mathrm{D}_{6}$
type);

  3. (3)

“minus lines” full rank non-split subspaces, e.g., $q(x,y)=x^{2}+xy+y^{2}$ –
$1120=28\cdot 120/3$ elements ($\mathrm{A}_{2}+\mathrm{E}_{6}$ type);

  4. (4)

full rank split subspaces, e.g., $q(x,y)=xy$ – $4320$ elements.

As a check, the Grassmannian $\operatorname{Gr}(2,8)$ has Betti numbers

$1\quad 1\quad 2\quad 2\quad 3\quad 3\quad 4\quad 3\quad 3\quad 2\quad 2\quad
1\quad 1$

and thus, by the Weil conjectures, $10795$ points of $\mathbb{F}_{2}$. Note
that

$10795=1575+3780+1120+4320.$

### What about arbitrary rank?

Let $\mathrm{A}_{1}$ and $\mathrm{A}_{2}$ be indefinite lattices of rank
$r\geq 2$ in the same genus. Consider overlattices

$\mathrm{L}_{1}\supset\mathrm{A}_{1}\oplus\mathrm{E}_{8}(-2),\quad\mathrm{L}_{2}\supset\mathrm{A}_{1}\oplus\mathrm{E}_{8}(-2)$

associated with subspaces $H\subset e_{8}(-2)$ in the same orbit, so we have
$d(\mathrm{L}_{1})\simeq d(\mathrm{L}_{2})$. It follows that
$\mathrm{L}_{1}\simeq\mathrm{L}_{2}$ provided the $d(\mathrm{L}_{i})$ have
rank at most $11$ (see Proposition 20); this holds for Picard lattices of $K3$
surfaces. Assuming $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ arise as Picard
lattices of K3 surfaces, we obtain results as in Propositions 21 and 24.

We conclude with one last observation:

######  Proposition 25.

The existence of a Nikulin structure for one member of a derived equivalence
class induces Nikulin structures on all K3 surfaces in the equivalence class.

Suppose $X_{1}$ and $X_{2}$ are derived equivalent and $X_{1}$ admits a
Nikulin involution. Proposition 20 implies

$\operatorname{Pic}(X_{1})\simeq\operatorname{Pic}(X_{2})$

and we obtain a copy of $\mathrm{E}_{8}(-2)\subset\operatorname{Pic}(X_{2})$.
Proposition 19 guarantees $X_{2}$ admits a Nikulin involution as well.

##  9\. Geometric application

In this section, we present a geometric application of the study of Nikulin
involutions, up to derived equivalence.

Let $(X_{1},f_{1})$ and $(X_{2},f_{2})$ denote derived equivalent K3 surfaces
of degree $12$, admitting Nikulin involutions $\iota_{j}:X_{j}\rightarrow
X_{j}$ with $\iota_{j}^{*}f_{j}=f_{j}$ for $j=1,2$. We assume Picard groups

$\operatorname{Pic}(X_{j})=\mathbb{Z}f_{j}\oplus\mathrm{E}_{8}(-2).$

Note that the derived equivalence induces natural identifications between the
$\mathrm{E}_{8}(-2)$ summands of $\operatorname{Pic}(X_{1})$ and
$\operatorname{Pic}(X_{2})$. In particular, we obtain bijections between the
fixed-point loci

$X^{\iota_{1}}_{1}=X^{\iota_{2}}_{2}.$

Let $Z_{j}\subset X_{j}$ denote triples of fixed points compatible with these
bijections. Assuming the $X_{j}$ are generic, i.e. defined by quadratic
equations in $\mathbb{P}^{7}$, the fixed points are not collinear.

Projection from the $Z_{j}$ gives surfaces

$\operatorname{Bl}_{Z_{j}}(X_{j})\rightarrow Y_{j}\subset\mathbb{P}^{4}$

where the blowup normalizes the image of the projection. These constructions
are compatible with the involutions on each side.

We claim that the construction of [HL18] gives a Cremona transform

$\phi:\mathbb{P}^{4}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\mathbb{P}^{4}$

such that

  * •

the indeterminacy of $\phi$ is $Y_{1}$;

  * •

the indeterminacy of $\phi^{-1}$ is $Y_{2}$;

  * •

$\phi$ is compatible with the involutions $\iota_{1}$ and $\iota_{2}$ induced
in the $\mathbb{P}^{4}$’s.

Indeed, the construction induces an isogeny of
$\mathrm{H}^{2}(X_{1},\mathbb{Z})$ and $\mathrm{H}^{2}(X_{2},\mathbb{Z})$
induced by $\phi$, restricting to an isomorphism of the primitive cohomology

$f_{1}^{\perp}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}f_{2}^{\perp}.$

The construction entails designating projection loci $Z^{\prime}_{j}\in
X_{j}^{[3]}$ compatible with the associated

$X_{1}^{[3]}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X_{2}^{[3]},$

our stipulation that the $Z_{j}$ consist of suitable fixed points gives
compatible projection loci.

Suppose that
$\phi:\mathbb{P}^{n}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}\mathbb{P}^{n}$
is birational and equivariant for the action of a finite group $G$. In this
case, [KT22, Thm. 1] introduces a well-defined invariant

(9.1) $C_{G}(\phi):=\sum_{\begin{subarray}{c}E\in\mathrm{Ex}_{G}(\phi^{-1})\\\
\mathrm{gen.stab}(E)=\\{1\\}\end{subarray}}[E\righttoleftarrow
G]-\sum_{\begin{subarray}{c}D\in\mathrm{Ex}_{G}(\phi)\\\
\mathrm{gen.stab}(D)=\\{1\\}\end{subarray}}[D\righttoleftarrow
G]\in\mathbb{Z}[\mathrm{Bir}_{G,n-1}],$

taking values in the free abelian group on $G$-birational isomorphism classes
of algebraic varieties of dimension $n-1$. In this case, the terms are the
projectivized normal bundles of $Y_{1}$ and $Y_{2}$, taken with opposite
signs. It is worth mentioning that the underlying K3 surfaces $X_{1}$ and
$X_{2}$ are isomorphic by Proposition 21, and the group actions are conjugate
under derived equivalences but not under automorphisms. The difference of
classes of exceptional loci in (9.1) is nonzero due to Proposition 26 below.
This gives an instance where the refinement of the invariant $c(\phi)$ in
[LSZ20], [LS22] using group actions yields new information.

######  Proposition 26 (cf. Thm. 2, [LS10]).

Let $X_{1}$ and $X_{2}$ be smooth projective $G$-varieties that are not
uniruled. Then any $G$-equivariant stable birational equivalence

$X_{1}\times\mathbb{P}^{r}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X_{2}\times\mathbb{P}^{s},$

with trivial $G$-action on the second factors, arises from a $G$-equivariant
birational equivalence

$X_{1}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X_{2}.$

###### Proof.

Our assumption – that $X_{1}$ and $X_{2}$ are not uniruled – means that

$X_{1}\times\mathbb{P}^{r}\rightarrow X_{1},\quad
X_{2}\times\mathbb{P}^{s}\rightarrow X_{2}$

are maximal rationally-connected (MRC) fibrations. Since
$X_{1}\times\mathbb{P}^{r}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X_{2}\times\mathbb{P}^{s}$,
the functoriality of MRC fibrations [Kol96, IV.5.5] gives a natural birational
map

$X_{1}\stackrel{{\scriptstyle\sim}}{{\dashrightarrow}}X_{2}.$

When the varieties admit $G$-actions, the induced birational map is compatible
with these actions. ∎

##  10\. Enriques involutions

Let $S$ be an Enriques surface over $\mathbb{C}$. Its universal cover is a K3
surface $X$ with covering involution $\epsilon:X\rightarrow X$, a fixed-point-
free automorphism of order two, called an Enriques involution.

The classification of Enriques surfaces $S$ up to derived equivalence boils
down to the classification of pairs $(X,\epsilon)$ up to $C_{2}$-equivariant
derived equivalence [BM01, §6] (and [BM17] more generally). Derived equivalent
Enriques surfaces are isomorphic [BM01, Prop. 6.1].

A number of authors have classified Enriques involutions on a given K3 surface
$X$, modulo its automorphisms $\operatorname{Aut}(X)$:

  * •

Kondo gave the first examples with finite $\operatorname{Aut}(S)$ [Kon92].

  * •

Ohashi showed that there finitely many $\operatorname{Aut}(X)$-orbits of such
involutions. In the Kummer case, the possible quotients are classified by
nontrivial elements of the discriminant group of the Néron-Severi group
$\mathrm{NS}(X)$. There are 15 on general Kummer surfaces of product type, 31
in a general Jacobian Kummer surface, but the number is generally unbounded
[Oha07], [Oha09].

  * •

Shimada and Veniani consider singular (i.e. rank $20$) K3 surfaces; one of
their results is a parametrization of $\operatorname{Aut}(X)$-orbits on the
set of Enriques involutions; the number of such orbits depends only on the
genus of the transcendental lattice $T(X)$ [SV20, Thm. 3.19].

These results are based on lattice theory: two Enriques involutions on a K3
surface $X$ are conjugate via $\operatorname{Aut}(X)$ if an only if the
corresponding Enriques quotients are isomorphic [Oha07, Prop. 2.1].

Let

$\mathrm{M}:=\mathrm{U}\oplus\mathrm{E}_{8}(-1)$

be the unique even unimodular hyperbolic lattice of rank 10; we have

$\operatorname{Pic}(S)/\mathrm{torsion}\simeq\mathrm{M}$

and

$\operatorname{Pic}(X)\supseteq\mathrm{M}(2)$

as a primitive sublattice. This coincides with the invariant sublattice

$\operatorname{Pic}(X)^{\epsilon=1}\subset\operatorname{Pic}(X)$

under the involution $\epsilon$. Let $\mathrm{N}$ denote the orthogonal
complement to $\mathrm{M}$ in $\mathrm{H}^{2}(X,\mathbb{Z})$, which coincides
with $\mathrm{H}^{2}(X,\mathbb{Z})^{\epsilon=-1}$; note that
$T(X)\subset\mathrm{N}$. We have

$\mathrm{N}\simeq\mathrm{U}\oplus\mathrm{U}(2)\oplus\mathrm{E}_{8}(-2)$

which has signature $(2,10)$. Thus

$\operatorname{Pic}(X)^{\epsilon=-1}=T(X)^{\perp}\subset\mathrm{N}$

has negative definite intersection form. The following result gives a
criterion for the existence of Enriques involutions [Keu16, Thm. 1], [Oha07,
Thm. 2.2], [SV20, Thm. 3.1.1]:

######  Proposition 27.

Let $X$ be a K3 surface. Enriques involutions on $X$ correspond to the
following data: Primitive embeddings

$T(X)\subset\mathrm{\mathrm{N}}\subset\mathrm{H}^{2}(X,\mathbb{Z})$

such that the orthogonal complement to $T(X)$ in $\mathrm{N}$ does not contain
$(-2)$-classes.

In particular, let $X$ be a K3 surface with an Enriques involution. Then:

  * •

$\operatorname{rk}\operatorname{Pic}(X)\geq 10$,

  * •

if $\operatorname{rk}\operatorname{Pic}(X)=10$ then there is a unique such
involution,

  * •

if $\operatorname{rk}\operatorname{Pic}(X)=11$ then $\operatorname{Pic}(X)$ is
isomorphic to [Oha07, Prop. 3.5]

    * –

$\mathrm{U}(2)\oplus\mathrm{E}_{8}\oplus\langle-2n\rangle$, $n\geq 2$, or

    * –

$\mathrm{U}\oplus\mathrm{E}_{8}(2)\oplus\langle-4n\rangle$, $n\geq 1$.

######  Proposition 28.

Let $X$ and $Y$ be derived equivalent K3 surfaces. Assume that $X$ admits an
Enriques involution. Then $X$ is isomorphic to $Y$. In particular, the
existence of an Enriques involution is a derived invariant.

###### Proof.

In Picard rank $\geq 12$, derived equivalence implies isomorphism. If $X$ and
$Y$ and derived equivalent of rank $10$ and $X$ admits an Enriques involution,
then $T(X)\simeq T(Y)$ and $\operatorname{Pic}(X)$ and $\operatorname{Pic}(Y)$
are stably isomorphic. In Picard ranks 10 and 11, it suffices to show that the
lattice $\mathrm{M}(2)$ is unique in its genus and all automorphisms of the
discriminant group $(d(\mathrm{M}(2)),q_{\mathrm{M}(2)}))$ lift to
automorphisms of $\mathrm{M}(2)$. This is implied by [Nik79b, Thm. 1.14.2].
Indeed, [SV20, Lem. 3.1.7] shows that $\operatorname{Pic}(X)$ satisfies these
two conditions whenever $X$ admits an Enriques involution. ∎

Corollary 11 implies (cf. [BM01, §6]):

######  Proposition 29.

Any $C_{2}$-equivariant derived autoequivalence

$(X,\epsilon_{1})\sim(X,\epsilon_{2})$

arises from an automorphism of $X$.

We observe a corollary of Proposition 2: Let $(X_{1},\epsilon_{1})$ and
$(X_{2},\epsilon_{2})$ denote K3 surfaces with Enriques involutions. They are
orientation reversing (i.e. skew) conjugate if

  * •

$\tau:T(X_{1})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}T(X_{2})$ as
lattices, with compatible Hodge structures;

  * •

$\operatorname{Pic}(X_{1})^{\epsilon_{1}=-1}$ and
$\operatorname{Pic}(X_{2})^{\epsilon_{2}=-1}$ have the same discriminant
quadratic form.

We explore this in more detail in the case of singular (rank $20$) K3
surfaces. The existence of involutions on singular K3 surfaces is governed by:

######  Proposition 30.

[Ser05] Let $X$ be a singular K3 surface with transcendental lattice $T(X)$ of
discriminant $d$. There is no Enriques involution on $X$ if and only if
$d\equiv 3\pmod{8}$ or

$T(X)=\begin{pmatrix}2&0\\\ 0&2\end{pmatrix},\quad\begin{pmatrix}2&0\\\
0&4\end{pmatrix},\text{ or }\begin{pmatrix}2&0\\\ 0&8\end{pmatrix}.$

The “most algebraic example”, i.e. the smallest discriminant admitting an
Enriques involution, has

$T(X)\simeq\left(\begin{matrix}2&1\\\ 1&4\end{matrix}\right).$

In this situation there are two possibilities. We write the maximal
sublattices

$\mathrm{N}\subset\operatorname{Pic}(X)$

such that the involution $\epsilon$ acts via $-1$.

We follow the notation [SV20, Table 3.1]. We consider lattices

$N^{144}_{10,7}(2),\quad N^{242}_{10,7}(2)$

where

$N^{242}_{10,7}(-1)\simeq\left(\begin{matrix}2&1\\\
1&4\end{matrix}\right)\oplus\mathrm{E}_{8}$

with $\mathrm{E}_{8}$ positive definite and

$N^{144}_{10,7}(2)(-1)\simeq\left(\begin{matrix}2&1&1&0&1&0&0&0&0&0\\\
1&2&0&0&0&0&0&0&0&0\\\ 1&0&2&1&0&0&0&0&0&0\\\ 0&0&1&4&0&0&0&0&0&0\\\
1&0&0&0&2&1&0&0&0&0\\\ 0&0&0&0&1&2&1&0&0&0\\\ 0&0&0&0&0&1&2&1&0&0\\\
0&0&0&0&0&0&1&2&1&0\\\ 0&0&0&0&0&0&0&1&2&1\\\
0&0&0&0&0&0&0&0&1&2\end{matrix}\right).$

According to magma, these two lattices are inequivalent but are in the same
spinor genus thus are stably equivalent.

These involutions are not derived equivalent. Indeed, passing to Mukai
lattices adds a hyperbolic summand $\mathrm{U}$ on which the involution acts
trivially. However, in the case at hand we are stabilizing the
$(-1)$-eigenspace. Thus these involutions are “skew equivalent” in the sense
of Section 6.

##  11\. Postscript on involutions in higher dimensions

There are many papers addressing the structure of involutions of higher-
dimensional irreducible holomorphic symplectic varieties.

  * •

Symplectic involutions of varieties of $K3^{[n]}$-type and their fixed loci
are classified in [KMO22].

  * •

For varieties of Kummer type – arising from an abelian surface $A$ –
involutions associated with $\pm 1$ on $A$ are analyzed in [HT13, Th. 4.4] and
[KMO22, Th. 1.3].

  * •

Anti-symplectic involutions on varieties of $K3^{[n]}$-type of degree two are
studied in [FMOS22].

  * •

Higher-dimensional analogs of Enriques involutions are studied in [OS11].

  * •

Involutions on cubic fourfolds – both symplectic (see [LZ22] and [HT10]) and
anti-symplectic – are studied in [Mar22]. The corresponding actions on
lattices are described explicitly.

  * •

Involutions on O’Grady type examples are considered in [MM22].

It is natural to consider whether derived equivalences of involutions on K3
surfaces $X_{1}$ and $X_{2}$ may be understood via equivalences of the induced
involutions on punctual Hilbert schemes and other moduli spaces.

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