Torelli Theorem for K3 surfaces and its proof. Period map. Moduli of polarized K3 surfaces (the case of Kummer surfaces). Formulation of Torelli Theorem for IHS manifolds.
The goal of this seminar is to present the fundamental tools for formulating questions related to the study of specific geometric properties of K3 surfaces (e.g., such as automorphisms, elliptic fibrations, Shioda-Inose structures). Furthermore, it aims to explore the analogue of these tools in the context of higher-dimensional varieties, known as irreducible holomorphic symplectic (IHS) manifolds or Hyperkähler.
1. Hirzebruch-Jung continued fractions
1.1. Basics
1.2. Wahl chains
1.3. Zero continued fractions
2. Singular and nonsingular algebraic surfaces
2.1. Generalities on surfaces and singularities
2.2. Cyclic quotient singularities
2.3. T-singularities
3. Deformations
3.1. General basic theory for affine and proper varieties
3.2. Q-Gorenstein deformations
3.3. Kollár–Shepherd-Barron correspondence
4. W-surfaces
4.1. Picard group, class group, and topology
4.2. MMP for W-surfaces I
4.3. MMP for W-surfaces II
5. N-resolutions
5.1. Existence and uniqueness
5.2. Braid group action
6. Exceptional collections of Hacking bundles
6.1. Hacking exceptional bundles
6.2. Hacking exceptional collections
6.3. Exceptional collections and H.e.c.s