Derived equivalences of K3 surfaces and orientation
TL;DR: In this paper, it was shown that every Fourier-Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing.
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Abstract: Every Fourier--Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms.
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•Book
Lectures on K3 Surfaces
Daniel Huybrechts
- 26 Sep 2016
TL;DR: Famous open conjectures, for example the conjectures of Calabi, Weil, and Artin–Tate, are discussed in general and for K3 surfaces in particular and each chapter ends with questions and open problems.
579
MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations
Arend Bayer,Emanuele Macrì +1 more
TL;DR: In this paper, the authors use wall-crossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X.
287
Hodge theory and derived categories of cubic fourfolds
TL;DR: In this paper, it was shown that Kuznetsov's cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.
Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes
TL;DR: In this paper, a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family is given, which is a product of Atiyah and Kodaira-Spencer classes.
157
Stability conditions for generic k3 categories
TL;DR: In this paper, the authors studied stability conditions on K3 categories and proved the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.
References
Deformation Quantization of Poisson Manifolds
TL;DR: In this paper, it was shown that every finite-dimensional Poisson manifold X admits a canonical deformation quantization, and that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the class of Poisson structures on X modulo diffeomorphisms.
Deformation quantization of Poisson manifolds, I
TL;DR: In this paper, it was shown that every finite-dimensional Poisson manifold X admits a canonical deformation quantization, which can be interpreted as correlators in topological open string theory.
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•Book
The geometry of moduli spaces of sheaves
Daniel Huybrechts,Manfred Lehn +1 more
- 01 Jan 1997
TL;DR: In this paper, the Grauert-Mullich Theorem is used to define a moduli space for sheaves on K-3 surfaces, and the restriction of sheaves to curves is discussed.
•Book
Fourier-Mukai transforms in algebraic geometry
Daniel Huybrechts
- 01 Jan 2006
TL;DR: In this paper, a quick tour of derived categories of coherent sheaves is presented, together with equivalence criteria for Fourier-Mukai transforms, and a canonical bundle is presented.
1K
Polynomial invariants for smooth four-manifolds
TL;DR: In this paper, the authors used infinite families of moduli spaces to define an infinite set of invariants for a simply connected 4-manifold X with b: odd and strictly greater than 1.
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