Related papers: Tropical Poincar\'e bundle, Fourier-Mukai transfor…

In complex K-theory, the Fourier-Mukai transform is an isomorphism between K-theory groups of a torus and its dual torus which is defined by pullback, tensoring by the Poincar\'e line bundle and pushforward. The Fourier-Mukai transform…

The classical Poincar\'e formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analogue of this formula. Along the way, we prove several…

The notion of a tropical vector bundle on a toric variety was recently introduced by Khan-Maclagan and Kaveh-Manon. In this paper, we study the Euler characteristic and rank of global sections for tropical vector bundles. We associate a…

In this paper I construct a geometric transformation for generalized 1-motives which extends the Fourier-Mukai transformation for O-Modules on abelian varieties, the geometric Fourier transformation for D-Modules on vector spaces and the…

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $\pi:M \to B,$ the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles $E$…

We introduce a sheaf-theoretic approach to tropical homology, especially for tropical homology with potentially non-compact supports. Our setup is suited to study the functorial properties of tropical homology, and we show that it behaves…

We introduce a notion of tropical vector bundle on a tropical toric variety which is a tropical analogue of a torus equivariant vector bundle on a toric variety. Alternatively it can be called a toric matroid bundle. We define equivariant…

We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…

We systematically develop a transform of the Fourier-Mukai type for sheaves on symplectic manifolds $X$ of any dimension fibred in Lagrangian tori. One obtains a bijective correspondence between unitary local systems supported on Lagrangian…

The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes.Pukhlikov and the second author noticed that the cohomology…

We construct a version of Fourier transform for families of real tori. This transform defines a functor from certain category associated with a symplectic family of tori to the category of holomorphic vector bundles on the dual family (the…

Let X be an abelian scheme over a scheme B. The Fourier--Mukai transform gives an equivalence between the derived category of X and the derived category of the dual abelian scheme. We partially extend this to certain schemes X over B (which…

The Baum-Connes map for finitely generated free abelian groups is a K-theoretic analogue of the Fourier-Mukai transform from algebraic geometry. We describe this K-theoretic transform in the language of topological correspondences, and…

We present an approach to Green-Lazarsfeld's generic vanishing combining gaussian maps and the Fourier-Mukai transform associated to the Poincar\`e line bundle. As an application we prove the Generic Vanishing Theorem for all normal…

This article is based on a talk given at the Kinosaki Symposium on Algebraic Geometry in 2015, about a work in progress. We describe a polarization on a derived equivalent abelian variety by using Fourier-Mukai theory. We explicitly…

In this work we define operator-valued Fourier transforms for suitable integrable elements with respect to the Plancherel weight of a (not necessarily Abelian) locally compact group. Our main result is a generalized version of the Fourier…

To prove log-concavity of the characteristic polynomials of matroids, Adiprasito-Huh-Katz proved the K\"{a}hler package (the hard Lefschetz theorem and the Hodge-Riemann bilinear relations) for their Chow rings. Amini-Piquerez generalized…

We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincar\'e duality for cohomology of tropical manifolds,…

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 construct the moduli spaces of tropical curves and tropical principally polarized abelian varieties, working in the category of (what we call) stacky fans. We define the tropical Torelli map between these two moduli spaces and we study…