K3 SURFACES WITH TWO INVOLUTIONS AND LOW PICARD NUMBER DINO FESTI WIM NIJGH DANIEL PLATT Abstract. LetXbe a complex algebraic K3 surface of degree 2dand with Picard number ρ.

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K3 SURFACES WITH TWO INVOLUTIONS AND LOW PICARD NUMBER

DINO FESTI, WIM NIJGH, DANIEL PLATT

Abstract.

Let

be a complex algebraic K3 surface of degree

and with Picard number

Assume that

admits two commuting involutions: one holomorphic and one anti-holomorphic.

In that case,

ρ≥1

when

d= 1

and

ρ≥2

when

d≥2

. For

d= 1

, the ﬁrst example deﬁned over

with

ρ= 1

was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by

Kond¯o, also deﬁned over

, can be used to realise the minimum

ρ= 2

for all

d≥2

. In these

notes we construct new explicit examples of K3 surfaces over the rational numbers realising the

minimum

ρ= 2

for

d= 2,3,4

. We also show that a nodal quartic surface can be used to realise

the minimum

ρ= 2

for inﬁnitely many diﬀerent values of

. Finally, we strengthen a result of

Morrison by showing that for any even lattice

of rank

1≤r≤10

and signature

(1, r −1)

there

exists a K3 surface Ydeﬁned over Rsuch that Pic YC= Pic Y∼

=N.

1. Introduction

One of the central objects of study in algebraic geometry are K3 surfaces, which beneﬁt of much

interest also from other areas of mathematics. In diﬀerential geometry, one landmark result is Yau’s

proof of the Calabi conjecture in [

] and, because of it, complex K3 surfaces are known to admit

metrics with holonomy equal to

(2). Through this connection with holonomy groups, K3 surfaces

often appear in the study of higher-dimensional manifolds with special holonomy.

One motivation for this article is the diﬀerential geometry of

-manifolds, i.e. the geometry

of manifolds of real dimension 7with holonomy equal to

. Here the group

denotes the

automorphism group of the octonions, see [

]. Only a few ways of constructing

-manifolds are

currently known, and one such way is the blowup procedure by Joyce and Karigiannis [

, Section

7.3], which uses a complex K3 surface

with two commuting involutions, i.e. bijective maps which

are their own inverse. More precisely, one needs a non-symplectic holomorphic involution

X→X

and an anti-holomorphic involution

X→X

. Another manifold construction by Kovalev and Lee

[

] uses a K3 surface with only one (non-symplectic holomorphic) involution. By using diﬀerent

models of K3 surfaces, such as quartics or double planes ramiﬁed over a sextic, one obtains many

examples of G2-manifolds.

Not much is known about which manifolds admit metrics with holonomy

, or how many such

metrics there are on a manifold that does. Because of this, much work in the ﬁeld is done on

constructing and studying examples. Using explicit examples of K3 surfaces, one obtains good

information about the metric on the resulting

-manifolds. In the future, this may be helpful

for studying possible metric degenerations of

-manifolds, similar to [

, Example 7.2]), which is

realised by two diﬀerent construction methods, exhibiting two diﬀerent metric degenerations. The

analogous question for K3 surfaces had likewise ﬁrst been studied via examples, but by now some

general results exist, see the references in Chen, Viaclovsky, and Zhang [

, Section 1] for an overview.

Date: February 13, 2024.

arXiv:2210.14623v2 [math.AG] 12 Feb 2024

2 DINO FESTI, WIM NIJGH, DANIEL PLATT

On the other hand, a program proposed by Donaldson and Thomas [

] to study the moduli space

-metrics suggests counting

-instantons, i.e. solutions to a certain partial diﬀerential equation

on principal bundles over the given manifold. If the manifold with holonomy

is obtained via

one of the two constructions above, one gets

-instantons from certain stable bundles over the

K3 surface, see Walpuski-Sá Earp [

], Walpuski [

], and the last author [

, Section 5.2]. Even

more than for

-manifolds, there are few general results about

-instantons and current work

concentrates on constructing and studying examples. The instanton constructions cited above use

detailed information about the K3 surfaces in question, so an ample supply of explicit examples

of K3 surfaces is helpful for these constructions. The main challenge in this research area is the

compactiﬁcation of the moduli space of

-instantons. As for metrics, it is hoped that explicit

examples of degenerations will suggest candidates for a compactiﬁcation.

As a general principle, checking whether a given bundle is stable becomes computationally harder

the more line bundles there are on the K3 surface. This can be seen in practice in the work of

Jardim, Menet, Prata and Sá Earp [13, Theorem 3].

The group of line bundles modulo isomorphism on a complex K3 surface

is called the Picard

group, and we will denote it by

Pic X

. It is a ﬁnitely generated free abelian group, and its rank,

which we denote by

(

), is called the Picard number of

. The Picard group can be endowed

with a non-degenerate bilinear form, making it a lattice (that is, a ﬁnitely generated free abelian

group with a non-degenerate bilinear form) called the Picard lattice. A lattice can be identiﬁed

by a matrix, called its Gram matrix, see Remark 2.3 for deﬁnition and notations. We are then

interested in examples of K3 surfaces with low Picard number and two commuting involutions, one

holomorphic and one non-holomorphic. What is the lowest Picard number possible for such K3

surfaces, for any given degree? Combining results of Kondo and Elsenhans–Jahnel, [

], we give

the following complete answer to this question.

Theorem 1.1. Let

be a K3 surface of degree 2

and with Picard rank

, admitting a holomorphic

and an anti-holomorphic involution which commute. If

= 1, then

ρ≥

1and there exist examples

deﬁned over

with

= 1. If

d >

1, then

ρ≥

2and there exist examples over

with

= 2 and

Picard lattice [0 1 0].

The proof of the theorem uses an explicit example of a K3 surface which can be endowed with

multiple polarisations. As mentioned above, it is desirable to have many more explicit examples for

applications in G2-geometry. We therefore give the following explicit examples.

Example 1.2. We present the following new examples:

(1)

in §6, a construction method for K3 surfaces over

of degree 4and 8with Picard lattice

[4 5 2], together with an explicit example;

(2)

in §7, a construction method for K3 surfaces over

of degree 6with Picard lattice [6 6 2],

together with an explicit example;

(3)

in §8, for every

d >

3, we show the existence of K3 surfaces over

with Picard lattice

[2 d+1 2d]and polarization of degree 2d.

We also review the following known K3 surface:

(4)

in §4, a K3 surface deﬁned over

with Picard lattice [4 0

−

2] admitting inﬁnitely many

polarizations of diﬀerent degrees.

We say that a complex K3 surface

can be deﬁned over a ﬁeld

admits a projective model,

whose deﬁning equations have coeﬃcients contained in

. The existence of an anti-holomorphic

involution on

is guaranteed as soon as the surface can be deﬁned over

: indeed, in this case,

the complex conjugation provides an anti-holomorphic involution. In the following, we will always

K3 SURFACES WITH TWO INVOLUTIONS AND LOW PICARD NUMBER 3

use this one as it will commute with the holomorphic involutions we will provide. Conversely, if

there exists an anti-holomorphic involution on

, then

can be deﬁned over

(see Silhol’s [

Proposition 1.3]).

A biholomorphic map on a complex K3 surface

comes from an automorphism on the associated

algebraic surface. A holomorphic involution on

will commute with the anti-holomorphic one if

and only if the associated automorphism on the model of

over

can be deﬁned over

. As a

conclusion, we have the following important remark.

Remark 1.3. A complex K3 surface has a commuting holomorphic and anti-holomorphic involution

if and only if the underlying algebraic K3 surface can be deﬁned over

and admits an automorphism

of order 2. This observation gives the main idea of this work: we look for K3 surfaces that can

be deﬁned over

admitting an ample divisor

with

= 2. This ample divisor will provide the

automorphism of order 2we are after, see Lemma 3.1.

The paper is structured as follows. In §2 we introduce K3 surfaces and their Picard lattice,

reviewing basic properties and results. In the same section, we summarise the known results about

involutions of K3 surfaces. In §3 we review branched double covers of

which play a central role

in the rest of the article and we prove Theorem 1.1. In §4 we study nodal quartics, serving as a

link between K3 surfaces of degree 2and higher degrees as they provide a construction of a K3

surface over

with Picard rank 2realising inﬁnitely many degrees. Smooth quartics admitting

a holomorphic involution are studied in §5, where we prove that there are inﬁnitely many non-

isomorphic smooth quartics over

with Picard number 2and admitting a holomorphic involution.

We also show that if a quartic surface has a holomorphic involution that can be extended to a linear

involution of its ambient space

, then its Picard number is higher. We construct examples of

K3 surfaces of degrees 2

, with Picard number 2, deﬁned over

, and admitting a holomorphic

involution, for 2

= 4

8in §6 and for 2

= 6 in §7. In §8 we study when complex K3 surfaces

can be deﬁned over the real numbers and prove that any even lattice

of rank 1

≤r≤

10 and

signature (1

, r −

1) can be realized as Picard lattice of a K3 surface deﬁned over

. We use this

result to show the existence of inﬁnitely many non-isomorphic K3 surfaces deﬁned over

with

= 2

admitting an involution. The

Magma

code pertaining to our explicit examples can be found online at

github.com/danielplatt/quartic-k3-with-involution.

Acknowledgments

We would like to thank Alex Degtyarev, Alice Garbagnati, Bert van Geemen, Ronald van Luijk,

Bartosz Naskr¸ecki, and Simon Salamon for many useful conversations on this topic. We also thank

the anonymous referee for their comments. The ﬁrst author was partially supported by the PRIN

grant PRIN202022AGARB_01 while in Milano.

2. Some background

In this section, we provide the necessary background about K3 surfaces. We mostly follow

Huybrecht [

] and Kondo [

]; for basic notions and results in algebraic geometry we refer to

Hartshorne [11] and Griﬃth [10]. As a start, we give a formal deﬁnition of a complex K3 surface.

By complex K3 surface we mean a compact connected complex manifold

of dimension two with

trivial canonical bundle and

(

X, OX

) = 0. It follows that

admits a holomorphic 2-form that is

nowhere vanishing and unique up to scaling. Examples of K3 surfaces are, among others, smooth

quartic surfaces in

, smooth intersections of a cubic and a quadric in

, smooth intersections of

three quadrics in P5, and double covers of P2ramiﬁed over a smooth curve of degree six.

4 DINO FESTI, WIM NIJGH, DANIEL PLATT

An algebraic K3 surface

over a ﬁeld

is a projective, geometrically integral and smooth

variety

over

such that its canonical bundle Ω

X/k

is isomorphic to

and with

(

X, OX

) = 0.

By the GAGA theorem [

], we have the following connection between algebraic and complex K3

surfaces: starting with an algebraic K3 surface

over

, where

is a subﬁeld of

, the complex

points

(

)have a natural structure of a complex K3 surface; as a converse, if one starts with a

projective complex K3 surface

, then there exists an algebraic K3 surface

X′

over

such that

X′

(

) =

. More speciﬁcally, a complex K3 surface comes from an algebraic K3 surface if and only

if it can be embedded in projective space.

Remark 2.1. There are complex K3 surfaces that are not projective and hence not algebraic (in

fact non-projective K3 surfaces are dense in the moduli space of complex K3 surfaces). Nikulin

shows that if a K3 surface has a non-symplectic automorphism of ﬁnite order, then it is projective,

see [

, Theorems 0.1a) and 3.1a)]. As in both Joyce–Karigiannis’ and Kovalev–Lee’s constructions

anon-symplectic involution is needed, non-projective K3 surface will not be considered in this paper.

Because of Remark 2.1, our attention will focus on algebraic complex K3 surfaces. Therefore,

unless stated otherwise, a complex K3 surface will be assumed to be algebraic. We will use both the

algebraic and manifold structure on the K3 surface.

2.1. The Picard lattice of a K3 surface. Let

be a ﬁeld and let

be an algebraic closure of

Let

denote an algebraic K3 surface over

. Throughout this paper, we will denote by

the base

change of

. If

is a subﬁeld of

, e.g.

, we use the notation

to denote the

base change of Xto C.

The Picard group of

, denoted

Pic X

, is the group of isomorphism classes of invertible sheaves

under the tensor product. It is known that the Picard group is isomorphic to

(

X, O∗

where

O∗

denotes the sheaf whose sections over an open set

are the units in the ring

(

There is also a natural isomorphism

Cl X∼

=Pic X

, where

Cl X

Div X/∼

denotes the divisor class

group modulo linear equivalence [11, Corollary II.6.16].

There is a unique symmetric bilinear pairing

Div Xk×Div Xk→Z

, sending any two divisors

C, D

to the integer

C.D

, such that if

and

are non-singular curves meeting transversally, the

number

C.D

is the number of points of

C∩D

; for a divisor

the number

D.D

is called the

self-intersection of

. The pairing depends only on the linear equivalence classes and so it descends

to a pairing

Pic Xk×Pic Xk→Z

[

, Theorem V.1.1]. We can identify

Pic X

as a subgroup of

Pic Xk, and this pairing on Pic Xkwill then induce a pairing on Pic X.

is a subﬁeld of

, then using the GAGA theorem (see [

]), there is a natural identiﬁcation

of the group

Pic XC

with the group of line bundles on a topological space and so this deﬁnition

is equivalent with the deﬁnition given in the introduction. If

is a subﬁeld of

, we deﬁne the

Picard number of

to be the Picard number of the associated complex surface as deﬁned in the

introduction, or equivalently, to be the rank of Pic XC.

Remark 2.2. Although we can identify

Pic X

as a subgroup of

Pic Xk

, it is not always the case

that they have the same rank. In the explicit examples we are constructing, we will prove that

Pic X

Pic XC

, but in general this is deﬁnitely not the case. To avoid any confusion, we will use

the deﬁnition Picard number only for complex K3 surfaces.

Alattice Λis a ﬁnitely generated free abelian group together with a non-degenerate bilinear

pairing Λ

→Z

. The Picard group of a projective K3 surface is ﬁnitely generated and torsion

free [

, §1.2], and the induced intersection pairing is non-degenerate [

, Proposition 1.2.4], hence

making the group

Pic X

a lattice. We refer to this as the Picard lattice of

. If

is the class of a

K3 SURFACES WITH TWO INVOLUTIONS AND LOW PICARD NUMBER 5

curve of arithmetic genus

, the adjunction formula shows that

= 2

pa−

2. It follows that

Pic X

is an even lattice, i.e., D2is even for all Din Pic X.

Remark 2.3. Choosing a basis for

Pic X

, we can associate a Gram matrix to the bilinear pairing.

We introduce the notation [

a b c

]with

a, b, c ∈Z

for a lattice of rank 2 and a basis with Gram

matrix equal to

a b

b c

. We also introduce the notation

⟨a⟩

for a lattice of rank 1 and a basis with

Gram matrix equal to a.

2.2. The second cohomology group and the Hodge decomposition. Let

denote a complex

K3 surface. Using the exponential sequence

0→Z2πi

−−→ OX(C)

exp

−−→ O∗

X(C)→0

we get an induced injective map from the Picard group of a complex K3 surface

(

X, Z

denoted

c1: Pic X →H2

(

X, Z

). The cup product gives

(

X, Z

)the structure of a lattice, of which

the restriction to the Picard group is exactly the intersection pairing.

It is known that the lattice H2(X, Z)is isomorphic to the K3 lattice

ΛK3:= U⊕3⊕E8(−1)⊕2,

where

is the hyperbolic rank 2lattice [0 1 0], and

(

−

1) is the unique even unimodular negative-

deﬁnite lattice of rank 8[

, Proposition 1.35]. This is a lattice of signature (3

19). By the Hodge

index theorem [12, Subsection 1.2.2], the signature of the Picard lattice is (1, ρ(X)−1). It follows

that ρ(X)≤20.

The isomorphism between the spaces

(

X, Z

)and Λ

is not canonical. Choosing an isometry

αX:H2

(

X, Z

)

→

is called a marking of

, and we call the tuple (

X, αX

)amarked K3 surface.

For the remainder of this subsection, we will always assume that Xis marked by a marking αX.

We have a natural identiﬁcation

(

X, Z

)

⊗C∼

=H2

(

X, C

). As a complex K3 surface is a Kähler

manifold, there is a Hodge decomposition

H2(X, C) = M

p+q=2

Hp+q(X),

where

H1,1

(

)can be deﬁned over

and

H2,0

(

)and

H0,2

(

)are complex conjugates of each

other. Moreover, there are natural isomorphisms

Hp,q

(

)

∼

=Hq

(

Ω

). By the Lefschetz theorem

on (1

1)-classes [

, p.163], we have that the image of the map

equals the set

H1,1

(

)

∩H2

(

X, Z

Next we ﬁx a nowhere vanishing holomorphic 2-form

ωX∈H0

(

Ω

)

⊂H2

(

X, C

). As

H0(X, Ω2

X)is 1-dimensional by deﬁnition of a K3 surface, we have

H0(X, Ω2

X) = ⟨ωX⟩∼

=C.

Extending the cup product to H2(X, C), the form ωXsatisﬁes the Riemann conditions

(1) ω2

X= 0 and ωX·ωX>0,

see [

, p.57]. Observe that

H2,0⊕H0,2

(

) =

⟨Re

(

ωX

)

,Im

(

ωX

)

⟩

and

H1,1

(

)will be orthogonal

to this subspace. In particular, ωXdetermines the Hodge decomposition of H2(X, C).

We deﬁne the transcendental lattice of

, denoted

, to be the lattice (

Pic X

)

⊥⊆H2

(

X, Z

). By

the Lefschetz theorem on (1

1)-classes and the construction above, one can deduce that the 2-form

ωX

is contained in TX⊗C.

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K3SURFACESWITHTWOINVOLUTIONSANDLOWPICARDNUMBERDINOFESTI,WIMNIJGH,DANIELPLATTAbstract.LetXbeacomplexalgebraicK3surfaceofdegree2dandwithPicardnumberρ.AssumethatXadmitstwocommutinginvolutions:oneholomorphicandoneanti-holomorphic.Inthatcase,ρ≥1whend=1andρ≥2whend≥2.Ford=1,thefirstexampledefinedoverQwithρ...

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K3 SURFACES WITH TWO INVOLUTIONS AND LOW PICARD NUMBER DINO FESTI WIM NIJGH DANIEL PLATT Abstract. LetXbe a complex algebraic K3 surface of degree 2dand with Picard number ρ.

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