Max Lieblich

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We use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber’s theorem that they coincide for a separated union of two affine schemes), (2) give a new proof of de(More)
We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that(More)
We consider the stack of coherent algebras with proper support, a moduli problem generalizing Alexeev and Knutson’s stack of branchvarieties to the case of an Artin stack. The main results are proofs of the existence of Quot and Hom spaces in greater generality than is currently known and several applications to Alexeev and Knutson’s original construction:(More)
Given a proper morphism X → S, we show that a large class of objects in the derived category of X naturally form an Artin stack locally of finite presentation over S. This class includes S-flat coherent sheaves and, more generally, contains the collection of all S-flat objects which can appear in the heart of a reasonable sheaf of t-structures on X. In this(More)
We show that the number of deformation types of canonically polarized manifolds over an arbitrary variety with proper singular locus is finite, and that this number is uniformly bounded in any finite type family of base varieties. As a corollary we show that a direct generalization of the geometric version of Shafarevich’s original conjecture holds for(More)
We present a method for compactifying stacks of PGLn-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to show that various moduli spaces are irreducible and carry natural virtual fundamental classes. We also prove a version of the Skolem-Noether theorem for certain(More)
In this paper we describe several results, built up over the last decade or so, that use twisted sheaves to study questions in algebra, arithmetic, and cohomology. The paper is for the most part expository and aims to collect in one place what is known about these methods, their applications, and some questions about their future. It also hopes to be a(More)
CONTENTS 1. Introduction 1 2. Mukai motive 3 3. Kernels of Fourier-Mukai equivalences 10 4. Fourier-Mukai transforms and moduli of complexes 15 5. A Torelli theorem in the key of D 18 6. Every FM partner is a moduli space of sheaves 21 7. Finiteness results 22 8. Lifting kernels using the Mukai isocrystals 25 9. Zeta functions of FM partners over a finite(More)