Fourier-Mukai Transforms

The rank-one limit of the Fourier-Mukai transform

We give a formula for the specialization of the Fourier-Mukai transform on a semi-abelian variety of torus rank 1.

Some examples of isomorphisms induced by Fourier-Mukai functors

We give some examples of isomorphisms of moduli of sheaves induced by Fourier-Mukai functor. As applications, we give another proof on deformation type of some moduli spaces of sheaves on abelian and K3 surfaces.

Fourier-Mukai transforms and the wall-crossing behavior for Bridgeland's stability conditions

Bridgeland stability condition is preserved under the Fourier-Mukai transform by its definition. We explain the relation with Gieseker stability. By studying the wall-crossing behavior, we reprove that the moduli spaces of stable sheaves on abelian surfaces are birationally equivalent, if the associated Mukai vectors are related by isometries of the Mukai lattice.

Fourier-Mukai partners for very general special cubic fourfolds

We exhibit explicit examples of very general special cubic fourfolds with discriminant $d$ admitting an associated (twisted) K3 surface, which have non-isomorphic Fourier-Mukai partners. In particular, in the untwisted setting, we show that the number of Fourier-Mukai partners for a very general special cubic fourfold with discriminant $d$ and having an associated K3 surface, is equal to the number $m$ of Fourier-Mukai partners of its associated K3 surface, if $d \equiv 2 (\text{mod}\,6)$; else, if $d \equiv 0 (\text{mod}\,6)$, the cubic fourfold has $\lceil m/2 \rceil$ Fourier-Mukai partners.

Fourier-Mukai transforms of curves and principal polarizations

Given a Fourier-Mukai transform $Φ$ between the bounded derived categories of two smooth projective curves, we verifiy that the induced map between the Jacobian varieties preserves the principal polarization if and only if $Φ$ is an equivalence.

Fourier--Mukai partners of elliptic ruled surfaces

We study Fourier--Mukai partners of elliptic ruled surfaces. We also describe the autoequivalence group of the derived categories of ruled surfaces with an elliptic fibration, by using \cite{Ue15}.

关于三次超曲面的Kuznetsov分量的Fourier-Mukai型自等价的探讨

On Fourier-Mukai type autoequivalences of Kuznetsov components of cubic threefolds

本文确定了光滑复三次超曲面的Kuznetsov分量的所有Fourier-Mukai型自等价群，并为该范畴的Torelli定理的Fourier-Mukai版本提供了新证明。

Perverse coherent sheaves and Fourier-Mukai transforms on surfaces

We study perverse coherent sheaves on the resolution of rational double points. As examples, we consider rational double points on 2-dimensional moduli spaces of stable sheaves on K3 and elliptic surfaces. Then we show that perverse coherent sheaves appears in the theory of Fourier-Mukai transforms. As an application, we generalize the Fourier-Mukai duality for K3 surfaces to our situation.

非常一般特殊三次四维流形的K3范畴的Fourier-Mukai数

Fourier-Mukai numbers of K3 categories of very general special cubic fourfolds

本文针对非常一般特殊三次四维流形，研究其K3范畴的Fourier-Mukai伙伴数量，并给出了相应的计数公式。

The Fourier-Mukai transform of a universal family of stable vector bundles

In this note we prove that the Fourier-Mukai transform $Φ_{\mathcal{U}}$ induced by the universal family of the moduli space $\mathcal{M}_{\mathbb{P}^2}(4,1,3)$ is not fully faithful.

Fourier-Mukai transforms for quotient varieties

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 categories for Enriques and bielliptic surfaces.

Fourier-Mukai functors: a survey

This paper surveys some recent results about Fourier--Mukai functors. In particular, given an exact functor between the bounded derived categories of coherent sheaves on two smooth projective varieties, we deal with the question whether this functor is of Fourier--Mukai type. Several related questions are answered and many open problems are stated.

Fourier-Mukai transforms and canonical divisors

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 $κ(X)=dim X-1$.

Equivalences of triangulated categories and Fourier-Mukai transforms

We give a condition for an exact functor between triangulated categories to be an equivalence. Applications to Fourier-Mukai transforms are discussed. In particular, we obtain a large number of such transforms for K3 surfaces.

Non-uniqueness of Fourier-Mukai kernels

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 derived category of the product of two smooth projective schemes to the corresponding Fourier-Mukai functor.

由稳定层族给出的Fourier-Mukai变换与稳定性的保持

Fourier-Mukai transforms by a family of stable sheaves and preservation of stability

本文研究核为Gieseker稳定层族的Fourier-Mukai变换，给出保持层的Gieseker稳定性的条件。

由Fourier-Mukai变换诱导的K3曲面上稳定层模空间的双有理映射

Birational maps of moduli spaces of stable sheaves on a K3 surface induced by Fourier-Mukai transforms

该研究探讨由Fourier-Mukai变换诱导的皮卡数为1的K3曲面上稳定层模空间的双有理映射问题。

Categorical entropy for Fourier-Mukai transforms on generic abelian surfaces

In this note, we shall compute the categorical entropy of an autoequivalence on a generic abelian surface.

皮卡数为一的K3曲面的傅里叶-穆凯轨迹

Fourier-Mukai loci of K3 surfaces of Picard number one

本文刻画了皮卡数为一的复代数K3曲面导出范畴的傅里叶-穆凯轨迹，并证明其严格小于对应的Matsui谱。

A note on Fourier-Mukai transform

In this note, we consider the problem on the preservation of stability under the Fourier-Mukai transforms. We first show that the Fourier-Mukai transform on an abelian surface or a K3 surface does not always preserve the stability, even for a $μ$-stable vector bundle. We next provide some positive results on this problem. Finally we discuss the birational map of moduli spaces on an abelian surface.

Fully faithful Fourier-Mukai functors and generic vanishing

The aim of this mainly expository note is to point out that, given an Fourier-Mukai functor, the condition making it fully faithful is an instance of \emph{generic vanishing}. We test this point of view on some fairly classical examples, including the strong simplicity criterion of Bondal and Orlov, the standard flip and the Mukai flop.

Fourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic

We prove that a twisted Enriques (respectively, untwisted bielliptic) surface over an algebraically closed field of positive characteristic at least 3 (respectively, at least 5) has no non-trivial Fourier-Mukai partners.

*-Quantizations of Fourier-Mukai transforms

We study deformations of Fourier-Mukai transforms in general complex analytic settings. We start with two complex manifolds X and Y together with a coherent Fourier-Mukai kernel P on their product. Suppose that P implements an equivalence between the coherent derived categories of X and Y. Given an arbitrary formal quantization of X we construct a unique quantization of Y such that the Fourier-Mukai transform deforms to an equivalence of the derived categories of the quantizations. Here quantizations are understood in the framework of stacks of algebroids.

Fourier-Mukai transforms of line bundles on derived equivalent abelian varieties

We study the Fourier-Mukai functor D(Y) -> D(X) induced by the universal family on a fine moduli space Y for simple semihomogeneous vector bundles on an abelian variety X. The main result is that the Fourier-Mukai transform of a very negative line bundle on Y is ample if and only if the bundles parametrized by Y are nef.

Fourier-Mukai partners of singular genus one curves

The objective of the paper is to prove that, as it happens for smooth elliptic curves, any Fourier-Mukai partner of a projective reduced Gorenstein curve of genus one and trivial dualising sheaf, is isomorphic to itself.

Semi-homogeneous sheaves, Fourier-Mukai transforms and moduli of stable sheaves on abelian surfaces

This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier-Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier-Mukai transforms. As a consequence, an affirmative proof is given to the conjecture proposed by Mukai in the 1980s. This paper also includes an explicit description of the birational correspondence between the moduli spaces of stable sheaves and the Hilbert schemes.

Fourier-Mukai transformations on K3 surfaces with $ρ=1$ and Atkin-Lehner involutions

We show that there is a surjection from the Fourier-Mukai transformations on projective K3 surfaces with the Picard number $ρ(X)=1$ to so called to the group of Atkin-Lehner involutions. This was expected in Hosono-Lian-Oguiso-Yau's paper.

Fourier-Mukaï transform for $\mathcal{D}^{(0)}$-modules over formal abelian schemes

In 1996, Rothstein and Laumon simultaneously constructed a Fourier-Mukai transform for D-modules over a locally noetherian base of characteristic 0. This functor induces an equivalence of categories between quasi-coherent sheaves of D-modules over an abelian variety A and quasicoherent sheaves of O-modules over its universal vectorial extension A. In this article, we define a Fourier-Mukai transform for D-modules on an abelian formal scheme A/S = Spf (V), where V is a discrete valuation ring, and we discuss the extension of the classical results of Fourier-Mukai transform to this arithmetic ca…

A proof of the Trisecant Identity through the Fourier-Mukai transform

Using the technique of the Fourier-Mukai transform we give an explicit set of generators of the ideal defining an algebraic curve as a subscheme of its Jacobian. Essentially, these ideals are generated by the Fay's trisecant identities.

Torelli theorem via Fourier-Mukai transform

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

Lefschetz fixed point theorems for Fourier-Mukai functors and DG algebras

We propose some variants of Lefschetz fixed point theorem for Fourier-Mukai functors on a smooth projective algebraic variety. Independently we also suggest a similar theorem for endo-functors on the category of perfect modules over a smooth and proper DG algebra.

Counting Twisted Tame Fourier-Mukai Partners of an Ordinary K3 Surface

In this article, we prove that a tame twisted K3 surface over an algebraically closed field of positive characteristic has only finitely many tame twisted Fourier-Mukai partners and we give a counting formula in case we have an ordinary tame untwisted K3 surface. We also show that every tame twisted Fourier Mukai partner of a K3 surface of finite height is a moduli space of twisted sheaves over it.

Twisted Fourier-Mukai partners of Enriques surfaces

Bridgeland and Maciocia showed that a complex Enriques surface X has no Fourier-Mukai partners apart from itself: that is, if D^b(X) = D^b(Y) then X = Y. We extend this to twisted Fourier-Mukai partners: if alpha is the non-trivial element of Br(X) = Z/2 and D^b(X,alpha) = D^b(Y,beta), then X = Y and beta is non-trivial. Our main tools are twisted topological K-theory and twisted Mukai lattices.

Fourier-Mukai number of a K3 surface

We shall give a Counting Formula for the number of Fourier-Mukai partners of a K3 surface and consider three applications.

Adjoints to a Fourier-Mukai transform

Given a Fourier-Mukai functor $Φ$ in the general setting of singular schemes, under various hypotheses we provide both left and a right adjoints to $Φ$, and also give explicit formulas for them. These formulas are simple and natural, and recover the usual formulas when the Fourier-Mukai kernel is a perfect complex. This extends previous work of Anno and Logvinenko, and Hernandez Ruiperez, Lopez Martin and Sancho de Salas, and has applications to the twist autoequivalences of Donovan and Wemyss.

An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves

Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. In this paper we show that this result is false without the full faithfulness hypothesis.

The Fourier-Mukai transform made easy

We propose a slightly modified definition for the Fourier-Mukai transform (on abelian varieties) that makes it much easier to remember various formulas. As an application, we give relatively short proofs for two important theorems: the characterization of GV-sheaves in terms of vanishing, due to Hacon; and fact that M-regularity implies (continuous) global generation, due to Pareschi and Popa.

Deformations and Fourier-Mukai transforms

The aim of this paper is twofold: First we give an explicit construction of the infinitesimal deformations of the category Coh(X) of coherent sheaves on a smooth projective variety X. Secondly we show that any Fourier-Mukai transform $/Phi/colon D^b(X)/to D^b(Y)$ extends to an equivalence between the derived categories of the deformed Abelian categories.

A_{infinity}-structures, Brill-Noether loci and the Fourier-Mukai transform

We use A_{infinity}-formalism to study variation of cohomology spaces under formal deformations of coherent sheaves on projective varieties. As an application we describe formal neighborhoods of twisted Brill-Noether loci at some points. Another application is a computation of the Fourier-Mukai transform of some natural line bundles on symmetric powers of a curve.

Twisted Fourier-Mukai transforms for holomorphic symplectic fourfolds

We apply the methods of C{ă}ld{ă}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 certain four-fold and the derived category of twisted sheaves on its `mirror' partner. As corollaries, we show that the two spaces are connected by a one-parameter family of deformations through Lagrangian fibrations, and we extend the original Fourier-Mukai transform to degenerations of abelian surfaces.

Fourier-Mukai变换下稳定性的保持：核为Poincaré线丛

Preservation of stability under the Fourier-Mukai transform whose kernel is the Poincare line bundle

研究核为Poincaré线丛的Fourier-Mukai变换在任意阿贝尔曲面上保持层Gieseker稳定性的条件，并给出弱Brill-Noether性质的注记。

三次四维簇的Fourier-Mukai伙伴计数

Counting Fourier-Mukai partners of cubic fourfolds

本文开发算法计算三次四维簇的虚拟Fourier-Mukai伙伴数，证明温和假设下虚拟计数可恢复实际计数，应用于带辛自同构的三次簇得相关结论，证明辛自同构非Fourier-Mukai不变量。

Fourier-Mukai partners of canonical covers of bielliptic and Enriques surfaces

We prove that the canonical cover of an Enriques surface does not admit non-trivial Fourier-Mukai partners. We also show that the canonical cover of a bielliptic surface has at most one non-isomorphic Fourier-Mukai partner. The first result is then applied to birational Hilbert schemes of points and the second to birational generalised Kummer varieties. An appendix establishes that there are no exceptional or spherical objects in the derived category of a bielliptic surface.

An example of Fourier--Mukai partners of minimal elliptic surfaces

Let $X$ and $Y$ be smooth projective varieties over $\C$. We say that $X$ and $Y$ are \emph{D-equivalent} (or, $X$ is a \emph{Fourier--Mukai partner} of $Y$) if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories. The aim of this short note is to find an example of mutually D-equivalent but not isomorphic relatively minimal elliptic surfaces.

Relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano fibrations

We study the group of relative Fourier-Mukai transforms for Weierstrass fibrations, abelian schemes and Fano or anti-Fano fibrations. For Weierstrass and Fano or anti-Fano fibrations we are able to describe this group completely. For abelian schemes over an arbitrary base we prove that if two of them are relative Fourier-Mukai partners then there is an isometric isomorphism between the fibre products of each of them and its dual abelian scheme. If the base is normal and the slope map is surjective we show that these two conditions are equivalent. Moreover in this situation we completely determ…

Fourier-Mukai transform in the quantized setting

We prove that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves.

相对 Fourier-Mukai 伙伴的可数性

Countability of relative Fourier-Mukai partners

本文将 Fourier-Mukai 伙伴的可数性结果从代数闭域上的光滑真代数簇推广到诺特基概形上的光滑真概形，证明其相对 Fourier-Mukai 伙伴在同构意义下至多可数。

Stability Conditions Under the Fourier-Mukai Transforms on Abelian Threefolds

We realize explicit symmetries of Bridgeland stability conditions on any abelian threefold given by Fourier-Mukai transforms. In particular, we extend the previous joint work with Maciocia to study the slope and tilt stabilities of sheaves and complexes under the Fourier-Mukai transforms, and then to show that certain Fourier-Mukai transforms give equivalences of the stability condition hearts of bounded t-structures which are double tilts of coherent sheaves. Consequently, we show that the conjectural construction proposed by Bayer, Macri and Toda gives rise to Bridgeland stability conditions…

The group of autoequivalences and the Fourier-Mukai number of a projective manifold

Let $X$ be a smooth projective variety and $\Aut (D(X))$ the group of autoequivalences of the derived category of $X$. In this paper we show that $X$ has no Fourier-Mukai partner other than $X$ when $\Aut (D(X))$ is generated by shifts, automorphisms and tensor products of line bundles.

简单翻转的后代与傅里叶-穆卡等价性

Descendant and Fourier-Mukai equivalences for simple flops

研究简单翻转诱导的后代Gromov-Witten理论之间对应关系及傅里叶-穆卡等价性的兼容性

On the existence of Fourier-Mukai kernels

A theorem by Orlov states that any equivalence between the bounded derived categories of coherent sheaves of two smooth projective varieties, X and Y, is isomorphic to a Fourier-Mukai transform with kernel in the bounded derived category of coherent sheaves of the product XxY. In the case of an exact functor which is not necessarily fully faithful, we compute some sheaves that play the role of the cohomology sheaves of the kernel, and that are isomorphic to the latter whenever an isomorphism to a Fourier-Mukai transform exists. We then exhibit a class of functors that are not full or faithful …

On a relative Fourier-Mukai transform on genus one fibrations

We study relative Fourier-Mukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and non-projective. Grothendieck duality is used to prove a skew-commutativity relation between this equivalence of categories and certain duality functors. We use our results to explicitly construct examples of semi-stable sheaves on degenerating families of elliptic curves.

任意特征域上椭圆直纹曲面的Fourier--Mukai伙伴

Fourier--Mukai partners of elliptic ruled surfaces over arbitrary characteristic fields

本文将复数域上椭圆直纹曲面Fourier--Mukai伙伴集合的显式描述推广至任意特征域，并在证明中获得Popa--Schnell猜想的部分证据。

Fourier-Mukai transforms and stable sheaves on Weierstrass elliptic surfaces

On a Weierstraß elliptic surface $X$, we define a `limit' of Bridgeland stability conditions, denoted as $Z^l$-stability, by moving the polarisation towards the fiber direction in the ample cone while keeping the volume of the polarisation fixed. We describe conditions under which a slope stable torsion-free sheaf is taken by a Fourier-Mukai transform to a $Z^l$-stable object, and describe a modification upon which a $Z^l$-semistable object is taken by the inverse Fourier-Mukai transform to a slope semistable torsion-free sheaf. We also study wall-crossing for Bridgeland stability, and show th…