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 [3].
4. If
$G$
is finite and acts symplectically then its order is at most 960 by a result of S. Mukai, [2]. All finite abelian groups acting simplectically on a
$K3$
surface where classified by V. Nikulin in the 80’s, there are 14 such groups, [3].
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 [1], [3].

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
[1] 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.
[2] Shigeru Mukai, Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), no. 1, 183–221. MR 958597
[3] V.V. Nikulin, Finite groups of automorphisms of Kählerian surfaces of type K3, Uspehi Mat. Nauk 31 (1976), no. 2(188), 223–224.