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- Max Lieblich
- 2006

This paper consists of two parts: in the first, we use the deformation theory of twisted sheaves on stacks to generalize results of de Jong and Saltman on the period-index problem for the Brauer group, yielding various new cases of the standard conjecture. In the second part, we use the geometric techniques of the first part to relate the Hasse principle… (More)

- Max Lieblich
- 2005

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)

- Melanie Eggers, Matchett Wood, Max Lieblich, Barbara Fantechi, Bhargav Bhatt, Manjul Thank You To +3 others
- 2009

The association of algebraic objects to forms has had many important applications in number theory. Gauss, over two centuries ago, studied quadratic rings and ideals associated to binary quadratic forms, and found that ideal classes of quadratic rings are exactly parametrized by equivalence classes of integral binary quadratic forms. Delone and Faddeev, in… (More)

- Max Lieblich
- 2004

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)

- Max Lieblich
- 2005

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 reprove 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)

- Max Lieblich
- 2006

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)

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian approximation for such algebraic spaces, generalizing earlier results in the case of schemes.

We show that the number of deformation types of canonically polarized man-ifolds 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)

- Max Lieblich
- 2005

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)

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)