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 2form 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 

Remark 0.3 1. Observe that if \[\sigma\] is a non–symplectic automorphism then \[\alpha(g)\] is a nroot 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 3space \[\mathbb{P}^3(\mathbb{C})\] : \[x_0^4+x_1^4+x_2^4+x_3^4=0.\] 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. 

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 

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 nonsymplectic automorphisms of prime order., Math. Z. 268 (2011), no. 12, 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. 