## K3 Surfaces

Alessandra Sarti
Laboratoire de Mathématiques et Applications
Université de Poitiers (France)

Exercises

The aim of this problem session is to discuss some properties of automorphisms of
$K3$
surfaces.
Definition 0.1 Let
$X$
be a
$K3$
surface. An automorphism of
$X$
is a biholomorphic map
$\sigma:X\to X.$
We say that
$\sigma$
has finite order if it exists
$n\in \mathbb{N}\setminus \{0\}$
such that
$\sigma^n={\rm Id}$
.
We denote by
${\rm Aut}(X)$
the group of automorphisms of
$X$
. We recall some properties. Let
$G$
be a finite subgroup of
${\rm Aut}(X)$
, then for any
$g\in G$
we have an induced action:
$g^{*}\omega_X=\alpha(g)\omega_X,\quad \alpha(g)\in \mathbb{C}^{*}=\mathbb{C}\setminus \{0\}$
where we denote by
$\omega_X$
the holomorphic 2-form such that
$H^{0}(\Omega_X^2)=\mathbb{C}_{\omega_X}$
. We get a representation (i.e. a homomorphism of groups):
$\alpha_G\to \mathbb{C}^{*},\ g\mapsto \alpha(g).$
Let
$G_N:={\rm ker}(\alpha)=\{g\in G|\ g^{*}(\omega_X)=\omega_X\},$
we have an exact sequence of groups:
$1\to G_N\to G\to^{\alpha} \mathbb{C}^{*}$
so that
$G/G_N$
is isomorphic to
${\rm Im}(\alpha)\subset \mathbb{C}^{*}$
is a finite subgroup hence it is cyclic.
Definition 0.2 Let
$\sigma\in {\rm Aut}(X)$
of finite order
$n$
. We say that
$\sigma$
is called
1.  symplectic if
$\sigma^{*}\omega_X=\omega_X,$
i.e.
$\sigma\in G_N$
.
2. non-symplectic if
$\sigma^{*}\omega_X=\alpha(\sigma)\omega_X$
, with
$\alpha(\sigma)\not=0.$
3. purely non-symplectic if
$\sigma^{*}\omega_X=\zeta_n\omega_X$
, with
$\zeta_n$
a primitive
$n$
-root of the unity.
Remark 0.3    1. Observe that if
$\sigma$
is a non–symplectic automorphism then
$\alpha(g)$
is a n-root of unity not necessarily primitive.
2. If
$G$
is a finite group acting purely non–symplectically then
$G_N=\{Id\}$
and
$G\cong Im(\alpha)$
is cyclic, more precisely
$G\cong \nu_N=\langle e^{\frac{2\pi i}{N}}\rangle$
where
$N$
is the order of
$G$
.
3. All the purely non–symplectic automorphisms acting on a
$K3$
surface are classified, their order is at most 66, see .
4. If
$G$
is finite and acts symplectically then its order is at most 960 by a result of S. Mukai, . All finite abelian groups acting simplectically on a
$K3$
surface where classified by V. Nikulin in the 80’s, there are 14 such groups, .
To understand the properties of automorphisms of
$K3$
surfaces it is important to understand their fixed locus. If
$\sigma\in {\rm Aut}(X)$
, we define
${\rm Fix}(\sigma)=\{p\in X|\ \sigma(p)=p\}$
the fixed locus of
$\sigma$
.

Exercice 1 (on the fixed locus of an automorphism). Let
$X_4$
be the quartic
$K3$
surface (Fermat quartic) defined by the equation in complex projective 3-space
$\mathbb{P}^3(\mathbb{C})$
$x_0^4+x_1^4+x_2^4+x_3^4=0.$
Consider the automorphism of
$\mathbb{P}^3(\mathbb{C})$
induced by the automorphism of
$\mathbb{C}^4$

$\Sigma_2:\mathbb{C}^4\to \mathbb{C}^4,\ (x_0, x_1, x_2, x_3)\mapsto (-x_0, -x_1, x_2, x_3)$
this is an involution (i.e. an automorphism of order 2) of
$\mathbb{P}^3(\mathbb{C})$
that leaves invariant the equation of
$X_4$
so that its restriction
$\sigma_2$
is an involution of
$X_4$
.
1. Show that
$\sigma_2$
acts symplectically on
$X_4$
2. Show that the fixed locus
${\rm Fix}(\sigma_2)$
consists of 8 isolated points.
In fact it is a not easy result that any finite order automorphism with 8 isolated fixed points is a symplectic involution see , .

Exercice 2 (more on the fixed locus). We consider now the following automorphism of order 3 of
$\mathbb{P}^3(\mathbb{C})$
induced by
$\Sigma_3:\mathbb{C}^4\to \mathbb{C}^4,\ (x_0,x_1, x_2, x_3)\mapsto (x_0, x_1, \zeta_3x_2, \zeta_3^2x_3)$
with
$\zeta_3\in \mathbb{C}$
a third primitive root of unity.
1.  Compute all the homogeneous polynomials of degree 4 in the variables
$x_0, x_1, x_2, x_3$
that are invariant under the automorphism
$\Sigma_3$
.
2. We call
$X'_4$
a
$K3$
quartic surface defined as the zero set of a polynomial of the previous question. Show that
$\Sigma_3$
restricts to a symplectic automorphism
$\sigma_3$
of
$X'_4$
and describe the fixed locus.

Exercice 3 (on the type of the fixed locus). Let
$X$
be a
$K3$
surface and
$\sigma\in {\rm Aut}(X)$
of finite order. Show that
1. If
$\sigma$
is symplectic then it has only isolated fixed points.
2. If
$\sigma$
is purely non–symplectic than it can have fixed curves and/or isolated fixed points.
3. If
$\sigma$
is a non–symplectic involution, can one give a more precise description of the fixed locus ?

Exercice 4 (on the quotient by an automorphism). Let
$\sigma$
be a symplectic involution on a
$K3$
surface
$X$
. We have shown in the Exercice 3 that this has only isolated fixed points. Show that the quotient surface
$Y:=X/\langle\sigma\rangle$
has a singularity of type
$A_1$
(i.e. a point that locally looks like the vertex of a cone:
$\{u^2=z\omega\}\subset \mathbb{C}^3$
) in the image of a point
$p\in X$
such that
$\sigma(p)=p$
.

Problem session: Thursday 8th of november 2018.

References
 M. Artebani, A. Sarti, and S. Taki, K3 surfaces with non-symplectic automorphisms of prime order., Math. Z. 268 (2011), no. 1-2, 507–533, with an appendix by Shigeyuki Kondo.
 Shigeru Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), no. 1, 183–221. MR 958597
 V.V. Nikulin, Finite groups of automorphisms of Kählerian surfaces of type K3, Uspehi Mat. Nauk 31 (1976), no. 2(188), 223–224.