On the Ubiquity of Twisted Sheaves

2013 ◽  
pp. 205-227 ◽  
Author(s):  
Max Lieblich
Keyword(s):  
Twisted Sheaves

scholarly journals NON-SYMPLECTIC INVOLUTIONS ON MANIFOLDS OF -TYPE

2020 ◽  
pp. 1-25
Author(s):  
CHIARA CAMERE ◽  
ALBERTO CATTANEO ◽  
ANDREA CATTANEO
Keyword(s):  
Moduli Spaces ◽  
Quadratic Forms ◽  
Symplectic Manifolds ◽  
Hilbert Schemes ◽  
Small Rank ◽  
Twisted Sheaves ◽  
Formula Type

We study irreducible holomorphic symplectic manifolds deformation equivalent to Hilbert schemes of points on a $K3$ surface and admitting a non-symplectic involution. We classify the possible discriminant quadratic forms of the invariant and coinvariant lattice for the action of the involution on cohomology and explicitly describe the lattices in the cases where the invariant lattice has small rank. We also give a modular description of all $d$ -dimensional families of manifolds of $K3^{[n]}$ -type with a non-symplectic involution for $d\geqslant 19$ and $n\leqslant 5$ and provide examples arising as moduli spaces of twisted sheaves on a $K3$ surface.

Where to Go from Here

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Mirror Symmetry ◽  
Derived Categories ◽  
Derived Category ◽  
Vector Spaces ◽  
Crepant Resolution ◽  
Hilbert Schemes ◽  
Mckay Correspondence ◽  
Homological Mirror Symmetry ◽  
Twisted Sheaves ◽  
A Finite Group

This chapter gives pointers for more advanced topics, which require prerequisites that are beyond standard introductions to algebraic geometry. The Mckay correspondence relates the equivariant-derived category of a variety endowed with the action of a finite group and the derived category of a crepant resolution of the quotient. This chapter gives the results from Bridgeland, King, and Reid for a special crepant resolution provided by Hilbert schemes and of Bezrukavnikov and Kaledin for symplectic vector spaces. A brief discussion of Kontsevich's homological mirror symmetry is included, as well as a discussion of stability conditions on triangulated categories. Twisted sheaves and their derived categories can be dealt with in a similar way, and some of the results in particular for K3 surfaces are presented.

scholarly journals The bigger Brauer group and twisted sheaves

2009 ◽  
Vol 322 (4) ◽  
pp. 1187-1195 ◽  
Author(s):  
Jochen Heinloth ◽  
Stefan Schröer
Keyword(s):  
Brauer Group ◽  
Twisted Sheaves

scholarly journals A reconstruction theorem for abelian categories of twisted sheaves

2016 ◽  
Vol 2016 (712) ◽  
Author(s):  
Benjamin Antieau
Keyword(s):  
Coherent Sheaves ◽  
Abelian Categories ◽  
Reconstruction Theorem ◽  
Twisted Sheaves

AbstractWe use an idea of Rosenberg to prove a reconstruction theorem for abelian categories of α-twisted quasi-coherent sheaves on quasi-compact and quasi-separated schemes

scholarly journals Verra Four-Folds, Twisted Sheaves, and the Last Involution

2018 ◽  
Vol 2019 (21) ◽  
pp. 6661-6710 ◽  
Author(s):  
Chiara Camere ◽  
Grzegorz Kapustka ◽  
Michał Kapustka ◽  
Giovanni Mongardi
Keyword(s):  
Moduli Spaces ◽  
Hilbert Scheme ◽  
K3 Surfaces ◽  
Symplectic Manifolds ◽  
K3 Surface ◽  
Enriques Surfaces ◽  
Twisted Sheaves

Abstract We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of K3[2]-type admitting non-symplectic involutions with invariant lattices U(2) ⊕ D4(−1) or U(2) ⊕ E8(−2). This complements the results obtained in [43], [13], and the results from [29] about the geometry of irreducible holomorphic symplectic (IHS) four-folds constructed using the Hilbert scheme of (1, 1) conics on Verra four-folds. As a byproduct we find that IHS four-folds of K3[2]-type with Picard lattice U(2) ⊕ E8(−2) naturally contain non-nodal Enriques surfaces.

scholarly journals Cohomological obstruction theory for Brauer classes and the period-index problem

2010 ◽  
Vol 8 (3) ◽  
pp. 419-435 ◽  
Author(s):  
Benjamin Antieau
Keyword(s):  
Positive Integer ◽  
Homotopy Theory ◽  
Cohomological Dimension ◽  
Stable Homotopy ◽  
Homotopy Groups ◽  
Stable Homotopy Groups ◽  
Open Subscheme ◽  
Finite Set ◽  
Twisted Sheaves ◽  
Set Of Points

AbstractLet U be a connected noetherian scheme of finite étale cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that α is a class in H2(Uét,ℂm)tors. For each positive integer m, the K-theory of α-twisted sheaves is used to identify obstructions to α being representable by an Azumaya algebra of rank m2. The étale index of α, denoted eti(α), is the least positive integer such that all the obstructions vanish. Let per(α) be the order of α in H2(Uét,ℂm)tors. Methods from stable homotopy theory give an upper bound on the étale index that depends on the period of α and the étale cohomological dimension of U; this bound is expressed in terms of the exponents of the stable homotopy groups of spheres and the exponents of the stable homotopy groups of B(ℤ/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then , where [] is the integer part of , whenever per(α) is divided neither by the characteristic of k nor by any primes that are small relative to d.

Moduli of twisted sheaves

2007 ◽  
Vol 138 (1) ◽  
pp. 23-118 ◽  
Author(s):  
Max Lieblich
Keyword(s):  
Twisted Sheaves
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