Related papers: Hodge theory of twisted derived categories and the…

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…

Conditional on the Lefschetz standard conjecture in degree 2, we prove that the index of a Brauer class on a smooth projective variety divides a fixed power of its period, uniformly in smooth families. In the other direction, we reinterpret…

We prove the period-index conjecture for unramified Brauer classes on abelian threefolds. To do so, we develop a theory of reduced Donaldson-Thomas invariants for 3-dimensional Calabi-Yau categories, with the feature that the noncommutative…

We propose a geometric and categorical approach to the Hodge Conjecture for all smooth projective complex varieties. By embedding any such variety into a flat family with general fibers smooth complete intersections, we prove the conjecture…

We study the algebraic structure of the automorphism group of the derived category of coherent sheaves on a smooth projective variety twisted by a Brauer class. Our main results generalize results of Rouquier in the untwisted case.

In this paper we introduce two new ways to split ramification of Brauer classes on surfaces using stacks. Each splitting method gives rise to a new moduli space of twisted stacky vector bundles. By studying the structure of these spaces we…

We formulate a version of the integral Hodge conjecture for categories, prove the conjecture for two-dimensional Calabi-Yau categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and use…

We use twisted relative Picard varieties to split Brauer classes on projective varieties over algebraically closed fields by torsors for a fixed abelian scheme independent of the Brauer class. The construction is also used to prove that the…

We use twisted sheaves to study the problem of index reduction for Brauer classes. In general terms, this problem may be phrased as follows: given a field $k$, a $k$-variety $X$, and a class $\alpha \in \Br(k)$, compute the index of the…

We produce refined index obstructions, generalizing recently constructed index obstructions due to de Jong and Perry, for topologically trivial Brauer classes on smooth and projective complex varieties. We show that our refined obstructions…

Given a smooth compact complex surface together with a holomorphic line bundle on it, using the theory of Hodge modules, we compute the twisted Hodge groups/numbers of Hilbert schemes (or Douady spaces) of points on the surface with values…

By comparing the Postnikov towers of the classifying spaces of projective unitary groups and the differentials in a twisted Atiyah-Hirzebruch spectral sequence, we deduce a lower bound on the topological index in terms of the period, and…

We introduce and solve a period-index problem for the Brauer group of a topological space. The period-index problem is to relate the order of a class in the Brauer group to the degrees of Azumaya algebras representing it. For any space of…

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-K\"ahler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived…

The standard period-index conjecture for the Brauer group of a field of transcendence degree 2 over a $p$-adic field predicts that the index divides the cube of the period. Using Gabber's theory of prime-to-$\ell$ alterations and the…

In this paper, we establish an innovative framework in logarithmic Hodge theory for toroidal varieties, introducing weighted toroidal structures and developing a systematic obstruction theory for Hodge classes. Building upon recent advances…

We develop a framework for describing vector bundles on $\mu_n$-gerbes over curves and illustrate the construction through two detailed examples. Using the interpretation of Brauer classes as obstructions to descending determinantal line…

In this paper we construct a tilting sheaf for Severi-Brauer Varieties and Involution Varieties. This sheaf relates the derived category of each variety to the derived category of modules over a ring whose semisimple component consists of…

We show that any orientation preserving Hodge isometry between the Hodge structures of two K3 surfaces X and X' twisted by Brauer classes $\alpha$ resp. $\alpha'$ can be lifted to a Fourier-Mukai equivalence between the derived categories…

We study cohomologies and Hodge theory for complex manifolds with twisted differentials. In particular, we get another cohomological obstruction for manifolds in class $\mathcal{C}$ of Fujiki. We give a Hodge-theoretical proof of the…