Related papers: The Fourier-Mukai transform made easy

We prove that for every positive integer $m$, there exist infinitely many simple abelian varieties over $\mathbb{F}_2$ of order $m$. The method is constructive, building on the work of Madan--Pal in the case $m=1$ to produce an explicit…

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of…

We will discuss the Fourier-Mukai partners of a given abelian variety. The first part of the note is to give some basic theory of Fourier-Mukai partners and semi-homogenous vector bundles, then we will discuss the case when the kernel of an…

Ongoing work in quantum information emphasises the need for a structural understanding of quantum speedups: in this work, we focus on the quantum Fourier transform and the structures in quantum theory that enable it. We elucidate a general…

We show that the Fourier transform on the Jacobian of a curve interchanges "$\delta$ functions" at the curve and the theta divisor. The Torelli theorem is an immediate consequence.

We completely describe all semi-stable torsion free sheaves of degree zero on nodal cubic curves using the technique of Fourier-Mukai transforms. The Fourier-Mukai images of such sheaves are torsion sheaves of finite length, which we…

We study Fourier-Mukai transforms for smooth projective varieties whose canonical bundles have finite order, and relate them to equivariant transforms on certain finite covering spaces. Our results lead to new equivalences of derived…

We apply the methods of C{\u{a}}ld{\u{a}}raru to construct a twisted Fourier-Mukai transform between a pair of holomorphic symplectic four-folds. More precisely, we obtain an equivalence between the derived category of coherent sheaves on a…

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…

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 give a moduli-theoretic proof of the classical theorem of Gabriel, stating that a scheme can be reconstructed from the abelian category of quasi-coherent sheaves over it. The methods employed are elementary and allow us to extend the…

We provide an analog of the Hilbert-Chow morphism for generalized discriminants.

We establish an exponential error term for the renewal theorem in the context of products of random matrices, which is surprising compared with classical abelian cases. A key tool is the Fourier decay of the Furstenberg measures on the…

Let $X$ be a smooth projective variety. We study a relationship between the derived category of $X$ and that of a canonical divisor. As an application, we will study Fourier-Mukai transforms when $\kappa (X)=dim X-1$.

We prove that the kernels of Fourier-Mukai functors are not unique in general. On the other hand we show that the cohomology sheaves of those kernels are unique. We also discuss several properties of the functor sending an object in the…

This paper deals with properties of filtrations on vector spaces indexed by partially ordered finitely generated abelian groups, which we call multifiltrations. We discuss the usual properties of filtrations, like exhaustivity and…

We survey and investigate some computational aspects of the Fourier-Mukai transform.

We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on…

For a simple flop $X\dashrightarrow X'$, we construct a correspondence between genus $0$ descendant Gromov-Witten theories of $X$ and $X'$. We show that the Fourier-Mukai equivalence induced by $X\dashrightarrow X'$ is compatible, in a…

In this paper we show an alternative way of defining Fourier Series and Transform by using the concept of convolution with exponential signals. This approach has the advantage of simplifying proofs of transforms properties and, in our view,…