Related papers: Splitting unramified Brauer classes by abelian tor…

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…

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…

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.

We prove the existence of divided powers in \'etale Chow groups of abelian varieties over a separably closed field, and hence of an integral lift of the Fourier transform, away from the characteristic and up to $2$-torsion. The method is to…

We consider the Alexander polynomial of a plane algebraic curve twisted by a linear representation. We show that it divides the product of the polynomials of the singularity links, for unitary representations. Moreover, their quotient is…

We solve the period-index problem for the Brauer group of a general complex torus of dimension at least three, giving an explicit formula for the index of each Brauer class. As a consequence, the complex-analytic version of the period-index…

We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated to triangulated surfaces with arbitrarily oriented…

We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for…

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

For a finite braided tensor category we introduce its Picard crossed module consisting of the group of invertible module categories and the group of braided tensor autoequivalences. We describe the Picard crossed module in terms of braided…

We study the Brauer group of an abelian variety A over an algebraically closed field of characteristic p focusing on the p-primary torsion, the key part of which is a certain quasi-algebraic unipotent group U_A. We determine its dimension…

We prove that a finite braided tensor category A is invertible in the Morita 4-category BrTens of braided tensor categories if, and only if, it is non-degenerate. This includes the case of semisimple modular tensor categories, but also…

We introduce new techniques for working with presentations for a large class of (strict) tensor categories. We then apply the general theory to obtain presentations for partition, Brauer and Temperley-Lieb categories, as well as several…

We conjecture that every unramified Brauer class $\alpha\in \text{Br}(X)$ on a projective hyperk\"ahler manifold $X$ satisfies $\text{ind}(\alpha)\mid\text{per}(\alpha)^{\dim(X)/2}$. We provide evidence for this conjecture by proving it for…

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 show how to calculate the relative tensor product of bimodule categories (not necessarily invertible) using ladder string diagrams. As an illustrative example, we compute the Brauer-Picard ring for the fusion category…

This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics,…

We construct a braided structure on the algebra of K\"ahler differential forms of a commutative algebra twisted by an endomorphism. This generalises the construction done in M. Karoubi, Quantum Methods in Algebraic Topology, see…

The numerical invariants (global) cohomological length, (global) cohomological width, and (global) cohomological range of complexes (algebras) are introduced. Cohomological range leads to the concepts of derived bounded algebras and…

This article focuses on smooth, projective, and geometrically integral varieties $X$ defined over a number field $k$ with torsion-free geometric Picard groups. We establish an isomorphism between the Brauer groups of $X$ and its symmetric…