Fourier Mukai Transforms and Applications to String Theory
Bjorn Andreas, Daniel Hernandez Ruiperez
备注：
52 pages. To appear in Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. Minor changes, reference of conjecture in section 7.5 changed, references updated
发表信息：
Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat, 99 (1) (2005) 29-77
We give an introductory review of Fourier-Mukai transforms and their application to various aspects of moduli problems, string theory and mirror symmetry. We develop the necessary mathematical background for Fourier-Mukai transforms such as aspects of derived categories and integral functors as well as their relative version which becomes important for making precise the notion of fiberwise T-duality on elliptic Calabi-Yau threefolds. We discuss various applications of the Fourier-Mukai transform to D-branes on Calabi-Yau manifolds as well as homological mirror symmetry and the construction of vector bundles for heterotic string theory.
This paper studies the action of the Fourier-Mukai transform on moduli spaces of vertical torsion sheaves on elliptic Calabi-Yau threefolds in Weierstrass form. Moduli stacks of semistable one dimensional sheaves on such threefolds are identified with open and closed substacksof moduli stacks of vertical semistable two dimensional sheaves on their Fourier-Mukai duals. In particular, this yields explicit conjectural results for Donaldson-Thomas invariants of vertical two dimensional sheaves on K3-fibered elliptic Calabi-Yau threefolds.
We study the generalized hypergeometric system introduced by Gelfand, Kapranov and Zelevinsky and its relationship with the toric Deligne-Mumford (DM) stacks recently studied by Borisov, Chen and Smith. We construct series solutions with values in a combinatorial version of the Chen-Ruan (orbifold) cohomology and in the $K$-theory of the associated DM stacks. In the spirit of the homological mirror symmetry conjecture of Kontsevich, we show that the $K$-theory action of the Fourier-Mukai functors associated to basic toric birational maps of DM stacks are mirrored by analytic continuation trans…
We give a counting formula for the twisted Fourier-Mukai partners of a projective K3 surface. As an application, we describe all twisted Fourier-Mukai partners of a projective K3 surface of Picard number 1.
This is a survey article on mirror symmetry and Fourier-Mukai partners of Calabi-Yau threefolds with Picard number one based on recent works by the authors [HoTa1,2,3,4]. For completeness, mirror symmetry and Fourier-Mukai partners of K3 surfaces are also discussed.
The purpose of this note is twofold. We first review the theory of Fourier-Mukai partners together with the relevant part of Nikulin's theory of lattice embeddings via discriminants. Then we consider Fourier-Mukai partners of K3 surfaces in the presence of polarisations, in which case we prove a counting formula for the number of partners.
We use a relative Fourier-Mukai transform on elliptic K3 surfaces $X$ to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of $X$ reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on $X$ which, in view of the triviality of the quantum cohomology of K3 surfaces, can be interpreted as mirror symmetry.
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.
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.
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.
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.
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}.
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.
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.
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.
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.
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 $κ(X)=dim X-1$.
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.
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. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a B-field direction to a formal gerbe. We show these two deformations are Fourier-Mukai equivalent.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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…
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.
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.
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.
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.
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.
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.
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.
A generalization of the Fourier-Mukai transform is proposed. The construction is based on analogy with the classical picture of representations of the Heisenberg group.
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.
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.
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.
We compute a large number of moduli spaces of stable bundles on a general algebraic elliptic surface using a new class of Fourier-Mukai type transforms.
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.
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.
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.
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…
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.
