Graduate seminar K3 surfaces and their generalisations (Fall 2019)

Time and Place
Tuesdays 19:00-21:00, except  on public holidays in Conference Room 3 (2nd floor) of the Jin Chun Yuan West Building. 

Program and covered material
Yiming Zhong: The basic definitions of K3 surfaces and the line bundles.
    We will first talk about the basic definitions of K3 surfaces, including the definitions of algebraic and complex K3 surfaces and their classical invariants. We will also introduce the line bundles on K3 surfaces and their basic properties.

 Xiaoyu Su: Weil conjecture for K3 surfaces.
    We will introduce the Weil cohomology theory and state the Weil conjecture, in particular for K3 surface. Then we will sketch Deligne’s proof of the Weil conjecture for K3 surfaces.

 Xiaoyu Su: Kuga Satake construction and Deligne’s proof of K3 Weil conjecture.
     We introduce the Kuga-Satake construction, which associates to each weight 2 polarized Hodge structure of K3 type a weight one polarizable Hodge structure. We also explain how this enters Deligne’s proof of Weil conjecture for K3 surface using this construction.

Yiming Zhong: The moduli of K3 surfaces via Hilbert schemes.
    We introduce the construction of the moduli space of polarized K3 surfaces by means of Hilbert schemes. We first introduce the moduli functor and then construct the moduli space. Then we compute the dimension of the moduli space and show the automorphism group of a polarized K3 surface is finite. Finally we will briefly introduce the stack structure of the moduli space. This is a  general method for constructing a moduli space, hence it can be done in a great generality.

Yunpeng Zi: The moduli of K3 surfaces: period domains and period mappings I, II.
    We introduce  the relation between primitively polarized K3 surfaces and period domains. We begin with the definition of period domains and period mappings. We then introduce some famous theorems related to them, including the Local/Global Torelli theorem. Finally we construct the moduli of primitively polarized K3 surfaces from the moduli of marked K3 surfaces.  We will see the former is an arithmetic quotient of the latter.

Eduard Looijenga: Teichmueller spaces and Torelli theorems for hyperkaehler manifolds I, II, III.
     Kreck and Yang Su recently gave counterexamples to a version of the Torelli theorem for hyperkaehlerian manifolds as stated by Verbitsky. We extract the correct statement and to give a short proof of it. We also discuss several consequences, some of which may be new, while others are given new (shorter) proofs.

Xueqing Wen: Moduli spaces of stable sheaves on K3 surfaces I, II.
After  presenting some basic facts about the moduli spaces of stable sheaves on K3 surface, we try to prove that they are irreducible symplectic. Reference: Huybrechts, Daniel; Lehn, Manfred. The geometry of moduli spaces of sheaves.

Bin Wang
Nilpotent Cones and Sheaves on K3 surface.
    We will briefly explain a deformation between moduli of stable sheaves on K3 surfaces and moduli of Higgs bundles over curves. It is an application of the general theory which Xueqing talked about last time. If time permits, we will study the rank 2 case in more detail. 
R. Donagi, L. Ein, R.Lazarsfeld: Nilpotent cones and sheaves on K3 surfaces.
G. Laumon: Un analogue global du cone nilpotent.

Xun Lin: Derived category of coherent sheaves.
   Recently, Kawatani and Okwa constructed beautiful geometry constraints for existence of semi orthogonal decomposition of derived category. I will introduce their works. If time permits,  I will also talk about derived global torelli theorem of K3 surface.
Kawatani and Okawa:
Non-existence of semi orthogonal decompositions and  sections of the canonical bundle.
D.Huybrechts: Fourier-Mukai transforms in algebraic geometry.
D.Huybrechts: Lectures on K3 surfaces, Chapter 16.