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.