scholarly journals Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem

2016 ◽  
pp. 171-235 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
K3 Surfaces ◽  
Torelli Theorem

A New Proof of the Global Torelli Theorem for K3 Surfaces

1984 ◽  
Vol 120 (2) ◽  
pp. 237 ◽  
Author(s):  
Robert Friedman
Keyword(s):  
K3 Surfaces ◽  
Torelli Theorem

K3 Surfaces

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Moduli Spaces ◽  
Special Class ◽  
K3 Surfaces ◽  
Derived Categories ◽  
K3 Surface ◽  
Hodge Structures ◽  
Surface Theory ◽  
Torelli Theorem ◽  
Stable Sheaves ◽  
Moduli Theory

After abelian varieties, K3 surfaces are the second most interesting special class of varieties. These have a rich internal geometry and a highly interesting moduli theory. Paralleling the famous Torelli theorem, results from Mukai and Orlov show that two K3 surfaces have equivalent derived categories precisely when their cohomologies are isomorphic weighing two Hodge structures. Their techniques also give an almost complete description of the cohomological action of the group of autoequivalences of the derived category of a K3 surface. The basic definitions and fundamental facts from K3 surface theory are recalled. As moduli spaces of stable sheaves on K3 surfaces are crucial for the argument, a brief outline of their theory is presented.

scholarly journals GENERALIZED CALABI–YAU STRUCTURES, K3 SURFACES, AND B-FIELDS

2005 ◽  
Vol 16 (01) ◽  
pp. 13-36 ◽  
Author(s):  
DANIEL HUYBRECHTS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Symplectic Form ◽  
Weak Form ◽  
K3 Surfaces ◽  
Standard Theory ◽  
General Notion ◽  
Symplectic Structures ◽  
Special Cases ◽  
Torelli Theorem

Generalized Calabi–Yau structures, a notion recently introduced by Hitchin, are studied in the case of K3 surfaces. We show how they are related to the classical theory of K3 surfaces and to moduli spaces of certain SCFT as studied by Aspinwall and Morrison. It turns out that K3 surfaces and symplectic structures are both special cases of this general notion. The moduli space of generalized Calabi–Yau structures admits a canonical symplectic form with respect to which the moduli space of symplectic structures is Lagrangian. The standard theory of K3 surfaces implies surjectivity of the period map and a weak form of the Global Torelli theorem.

scholarly journals A motivic global Torelli theorem for isogenous K3 surfaces

2021 ◽  
Vol 383 ◽  
pp. 107674 ◽  
Author(s):  
Lie Fu ◽  
Charles Vial
Keyword(s):  
K3 Surfaces ◽  
Torelli Theorem

scholarly journals On the Period Map for Polarized Hyperkähler Fourfolds

2017 ◽  
Vol 2019 (22) ◽  
pp. 6887-6923 ◽  
Author(s):  
Olivier Debarre ◽  
Emanuele Macrì
Keyword(s):  
Moduli Space ◽  
K3 Surfaces ◽  
Finite Union ◽  
Picard Number ◽  
Fixed Degree ◽  
Period Map ◽  
Torelli Theorem ◽  
Open Embedding ◽  
Birational Automorphisms

Abstract We study smooth projective hyperkähler fourfolds that are deformations of Hilbert squares of K3 surfaces and are equipped with a polarization of fixed degree and divisibility. They are parametrized by a quasi-projective irreducible 20-dimensional moduli space and Verbitksy’s Torelli theorem implies that their period map is an open embedding. Our main result is that the complement of the image of the period map is a finite union of explicit Heegner divisors that we describe. We also prove that infinitely many Heegner divisors in a given period space have the property that their general points correspond to fourfolds which are isomorphic to Hilbert squares of a K3 surfaces, or to double EPW (Eisenbud–Popescu–Walter) sextics. In two appendices, we determine the groups of biregular or birational automorphisms of various projective hyperkähler fourfolds with Picard number 1 or 2.

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