Related papers: Twisted Derived Equivalences Between Abelian Varie…

An abelian stack is a stacky generalization of an abelian variety that was introduced by Brochard. Just as an abelian variety has a dual, an abelian stack $\mathcal{A}$ has a dual $\mathfrak{D}(\mathcal{A})$ which generalizes the classical…

We extend the usual projective Abel-Radon transform to the larger context of a smooth complete toric variety X. We define and study toric concavity attached to an algebraic splitting vector bundle on X and we prove a toric version of the…

We say that two abelian varieties $A$ and $A'$ defined over a field $F$ are polyquadratic twists if they are isogenous over a Galois extension of $F$ whose Galois group has exponent dividing $2$. Let $A$ and $A'$ be abelian varieties…

The classical Fourier-Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects.…

We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring $A:=\Bbbk[x_1,\ldots,x_n]$ is a graded twist of a unimodular Poisson…

An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of…

Braided non-commutative differential geometry is studied. In particular we investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided…

Let $k$ be an algebraic extension of $\mathbb F_p$ and $K/k$ a regular extension of fields (e.g. $\mathbb F_p(T)/\mathbb F_p$). Let $A$ be a $K$-abelian variety such that all the isogeny factors are neither isotrivial nor of $p$-rank zero.…

We prove that a bounded complex of coherent sheaves on an abelian variety A, whose Fourier-Mukai transform has support of dimension at most 1, extends to a perfect complex on the standard NC-smooth thickening of A and on the deformation…

Consider an absolutely simple abelian variety A defined over a number field K. For most places v of K, we study how the reduction A_v of A modulo v splits up to isogeny. Assuming the Mumford-Tate conjecture for A and possibly increasing K,…

In this paper we introduce the Fourier-Mukai transform for Lawson homology of abelian varieties and prove an inversion theorem for the Lawson homology as well as the morphic cohomology of abelian varieties. As applications, we obtain the…

Given two pure representations of the absolute Galois group of an $\ell$-adic number field with coefficients in $\overline{\mathbb{Q}}_p$ (with $\ell\neq p$), we show that the Frobenius-semisimplifications of the associated Weil--Deligne…

We introduce and study doubly twisted near-isometries. A doubly twisted near-isometry is a tuple of near-isometries satisfying certain relations determined by a prescribed family of unitaries, thereby generalizing the notion of doubly…

We study $p$-adic manifolds associated with twisted points of quotient stacks $\mathcal{X} = [U/G]$ and their quotient spaces $\pi:\mathcal{X} \to X$. We prove several structural results about the fibres of $\pi$ and derive in particular a…

Let $a_X:X\rightarrow \mathrm{Alb}\, X$ be the Albanese map of a smooth complex projective variety. Roughly speaking in this note we prove that for all $i \geq 0$ and $\alpha\in \mathrm{Pic}^0\, X$, the cohomology ranks $h^i(\mathrm{Alb}\,…

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 prove that the bounded derived category of coherent sheaves on a smooth projective complex variety reconstructs the isomorphism classes of fibrations onto smooth projective curves of genus $g\geq 2$. Moreover, in dimension at most four,…

Let $X$ be a Fano type variety and $(X,\Delta)$ be a log Calabi-Yau pair with $\Delta$ a Weil divisor. If $(X,\Delta)$ admits a polarized endomorphism, then we show that $(X,\Delta)$ is a finite quotient of a toric pair. Along the way, we…

This paper proves the following converse to a theorem of Mumford: Let $A$ be a principally polarized abelian variety of dimension five, whose theta divisor has a unique singular point, and suppose that the multiplicity of the singular point…

We prove that an $F$-isocrystal over an abelian variety defined over a perfect field of positive characteristic has constant slopes. This recovers and extends a theorem of Tsuzuki for abelian varieties over finite fields. Our proof exploits…