Fourier-Mukai partners of a K3 surface of Picard number one

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.

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.

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 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.

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。

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。

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 partners of abelian varieties

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 equivalence is given by a semi-homogenous vector bundle. In particular, given an abelian variety B, and a simple semi-homogenous vector bundle E on B, we will discuss for which E, it can be the image of the structure sheaf of the unit on A under some triangulated equivalence, where A is some abelian variety.

L等价与三次四维流形的傅里叶-穆卡伙伴

L-equivalence and Fourier--Mukai partners of cubic fourfolds

本文研究了三次四维流形的L等价与傅里叶-穆卡伙伴，通过衍生Torelli型准则和判别式条件，证明了计数公式并展示了有限的L等价类。

Twisted Fourier-Mukai number of a K3 surface

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.

Classification of Mukai pairs with dimension $4$ and rank $2$

We give the complete classification of Mukai pairs of dimension $4$ and rank $2$ with Picard number one, that is, pairs $(X,E)$ where $X$ is a Fano $4$-fold with Picard number one, and $E$ is an ample vector bundle of rank two on $X$ with $c_1(X) = c_1(E)$. Equivalently, the present paper completes the classification of ruled Fano $5$-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007.

Some remarks about the FM-partners of K3 surfaces with Picard numbers 1 and 2

In this paper we prove some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer $N$, there is a K3 surface with Picard number 2 and at least $N$ non-isomorphic FM-partners. We describe also the Mukai vectors of the moduli spaces associated to the Fourier-Mukai partners of K3 surfaces with Picard number 1.

非合冲三次四维簇的Fourier-Mukai伙伴与Gale对偶

Fourier-Mukai partners of non-syzygetic cubic fourfolds and Gale duality

本文研究非合冲三次四维簇，证明其恰有一个非平凡Fourier-Mukai伙伴，利用Gale对偶关联该伙伴，还构造相关反例探讨双有理性猜想。

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。

Unstable Lazarsfeld-Mukai bundles of rank 2 on a certain K3 surface of Picard number 2

Let $g$ and $c$ be any integers satisfying $g\geq3$ and $0\leq c\leq \lfloor\frac{g-1}{2}\rfloor$. It is known that there exists a polarized K3 surface $(X,H)$ such that $X$ is a K3 surface of Picard number 2, and $H$ is a very ample line bundle on $X$ of sectional genus $g$ and Clifford index $c$, by Johnsen and Knutsen([J-K] and [Kn]). In this paper, we give a necessary and sufficient condition for a Lazarsfeld-Mukai bundles of rank 2 associated with a smooth curve $C$ belonging to the linear system $|H|$ and a base point free pencil on $C$ not to be $H$-slope stable.

Some finiteness results for Fourier-Mukai partners

We develop some methods for studying the Fourier-Mukai partners of an algebraic variety. As applications we prove that abelian varieties have finitely many Fourier-Mukai partners and that they are uniquely determined by their derived category of coherent $D$-modules. We also generalize a famous theorem due to A. Bondal and D. Orlov.

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.

阿贝尔簇的真正 Fourier-Mukai 伙伴以及 Matsui 谱中 Fourier-Mukai 轨迹之外的点

Proper Fourier-Mukai partners of abelian varieties and points outside the Fourier-Mukai loci in Matsui spectra

本文通过分析导出范畴的 Matsui 谱，证明了阿贝尔簇的真正 Fourier-Mukai 伙伴仍为阿贝尔簇，并将结果扩展至正常概形；同时揭示了 Fourier-Mukai 轨迹之外的几何结构，为 Ito 猜想提供了基于高维简单阿贝尔簇的新反例。

Fourier-Mukai partners of a K3 surface and the cusps of its Kahler moduli

Using lattice theory, we establish a one-to-one correspondence between the set of Fourier-Mukai partners of a projective $K3$ surface and the set of 0-dimensional standard cusps of its Kahler moduli. We also study the relation between twisted Fourier-Mukai partners and general 0-dimensional cusps, and the relation between Fourier-Mukai partners with elliptic fibrations and certain 1-dimensional cusps.

Birational geometry of Beauville-Mukai systems II: general theory in low ranks

Via wall-crossing, we study the birational geometry of Beauville-Mukai systems on K3 surfaces with Picard rank one. We show that there is a class of walls which are always present in the movable cones of Beauville-Mukai systems. We give a complete description of the birational geometry of rank two Beauville-Mukai systems when the genus of the surface is small.

关于正特征域上阿贝尔簇的Fourier-Mukai伙伴的一个注记

A note on Fourier-Mukai partners of abelian varieties over positive characteristic fields

本文研究正特征域上阿贝尔簇的Fourier-Mukai伙伴，将Krug-Sosna关于Kummer栈的导出等价结果推广到奇特征情形，证明阿贝尔曲面导出等价与Kummer曲面同构等价，并建立超奇异阿贝尔簇的导出Torelli定理。

阿贝尔簇和K3曲面在正特征和混合特征下的傅里叶-穆卡伊伙伴

Fourier-Mukai partners of abelian varieties and K3 surfaces in positive and mixed characteristics

本文研究了在正特征和混合特征下阿贝尔簇和K3曲面的傅里叶-穆卡伊伙伴，证明了其提升性质并恢复了原有结果。

A remark on K3s of Todorov type (0,9) and (0,10)

Frequentely it happens that isogenous (in the sense of Mukai) K3 surfaces are partners of each other and sometimes they are even isomorphic. This is due, in some cases, to the (too high, e.g. bigger then or equal to 12) rank of the Picard lattice as showed by Mukai in [Muk3]. In other cases this is due to the structure of the Picard lattice and not only on its rank. This is the case, for example, of K3s with Picard lattice containing a latice of Todorov type (0,9) or (0,10)

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.

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。

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 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.

Mukai's Program (reconstructing a K3 surface from a curve) via wall-crossing, II

Let $C$ be a curve on a K3 surface $X$ with Picard group $\mathbb{Z}.[C]$. Mukai's program seeks to recover $X$ from $C$ by exhibiting it as a Fourier-Mukai partner to a Brill-Noether locus of vector bundles on $C$. We use wall-crossing in the space of Bridgeland stability conditions to prove this for genus $\ge14$. This paper deals with the case $g-1$ prime left over from Paper I.

On the Picard number of almost Fano threefolds with pseudo-index >1

We study Gorenstein almost Fano threefolds X with canonical singularities and pseudoindex > 1. We show that the maximal Picard number of X is 10 in general, 3 if X is Fano, and 8 if X is toric. Moreover, we characterize the boundary cases. In the Fano case, we prove that the generalized Mukai conjecture holds.

Rational curves and bounds on the Picard number of Fano manifolds

We prove that Generalized Mukai Conjecture holds for Fano manifolds $X$ of pseudoindex $i_X \ge (\dim X +3)/3$. We also give different proofs of the conjecture for Fano fourfolds and fivefolds.

Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds

We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their conjectural generalized Bogomolov-Gieseker inequality for certain tilt stable objects. We do this by proving that a suitable Fourier-Mukai transform preserves the heart of a particular conjectural stability condition. We also show that the only reflexive sheaves with zero first and second Chern classes are the flat line bundles.

向量丛模空间的Fourier-Mukai伙伴与自等价

Fourier-Mukai partners and autoequivalences of moduli spaces of vector bundles

本文分类了稳定向量丛模空间的Fourier-Mukai伙伴，并确定其导出自等价群，证明了亏格≥4时互素模空间导出等价当且仅当同构。

Fourier-Mukai partners of K3 surfaces in positive characteristic

We study Fourier-Mukai equivalence of K3 surfaces in positive characteristic and show that the classical results over the complex numbers all generalize. The key result is a positive-characteristic version of the Torelli theorem that uses the derived category in place of the Hodge structure on singular cohomology; this is proven by algebraizing formal lifts of Fourier-Mukai kernels to characteristic zero. As a consequence, any Shioda-supersingular K3 surface is uniquely determined up to isomorphism by its derived category of coherent sheaves. We also study different realizations of Mukai's Hod…

Birational geometry of Beauville-Mukai systems III: asymptotic behavior

Suppose that a Hilbert scheme of points on a K3 surface S of Picard rank one admits a rational Lagrangian fibration. We show that if the degree of the surface is sufficiently large compared to the number of points, then the Hilbert scheme is the unique hyperkähler manifold in its birational class. In particular, the Hilbert scheme is a Lagrangian fibration itself, which we realize as coming from a (twisted) Beauville-Mukai system on a Fourier-Mukai partner of S. We also show that when the degree of the surface is small our method can be used to find all birational models of the Hilbert scheme.

Preservation of semistability under Fourier-Mukai transforms

For a trivial elliptic fibration $X=C \times S$ with $C$ an elliptic curve and $S$ a projective K3 surface of Picard rank $1$, we study how various notions of stability behave under the Fourier-Mukai autoequivalence $Φ$ on $D^b(X)$, where $Φ$ is induced by the classical Fourier-Mukai autoequivalence on $D^b(C)$. We show that, under some restrictions on Chern classes, Gieseker semistability on coherent sheaves is preserved under $Φ$ when the polarisation is `fiber-like'. Moreover, for more general choices of Chern classes, Gieseker semistability under a `fiber-like' polarisation corresponds to …

Marked and labelled Gushel-Mukai fourfolds

We prove that the moduli stacks of marked and labelled Hodge-special Gushel-Mukai fourfolds are isomorphic. As an application, we construct rational maps from the stack of Hodge-special Gushel-Mukai fourfolds of discriminant $d$ to the moduli space of (twisted) degree-$d$ polarized K3 surfaces. We use these results to prove a counting formula for the number of 4-dimensional fibers of Fourier-Mukai partners of very general Hodge-special Gushel-Mukai fourfolds with associated K3 surface, and a lower bound for this number in the case of a twisted associated K3 surface.

3阶广义Kummer曲面上的Fourier–Mukai伙伴与广义Kummer结构

Fourier--Mukai partners and generalized Kummer structures on generalized Kummer surfaces of order $3$

本文研究3阶广义Kummer曲面与原Abel曲面的相互决定关系，按除子平方模9的情形刻画超越格、Hodge结构及Fourier–Mukai伙伴何时推出曲面同构。

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.

On the Picard number of Fano 6-folds with a non-small contraction

A generalization of S. Mukai's conjecture says that if $X$ is a Fano $n$-fold with Picard number $ρ_X$ and pseudo-index $i_X$, then $ρ_X(i_X-1) \leq n$, with equality if and only if $X \cong (\mathbb{P}^{i_X-1})^{ρ_X}$. In this paper, we prove that this conjecture holds if $n=6$ and either $X$ admits a contraction of fiber type or $X$ admits no small contractions.

Mukai's program (reconstructing a K3 surface from a curve) via wall-crossing

Let $C$ be a curve of genus $g \geq 11$ such that $g-1$ is a composite number. Suppose $C$ is on a K3 surface whose Picard group is generated by the curve class $[C]$. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai's program to this situation: we show how to reconstruct the K3 surface containing the curve $C$ as a Fourier-Mukai transform of a Brill-Noether locus of vector bundles on $C$.

Fourier-Mukai Transforms and Bridgeland Stability Conditions on Abelian Threefolds II

We show that the conjectural construction proposed by Bayer, Bertram, Macrí and Toda gives rise to Bridgeland stability conditions for a principally polarized abelian three-fold with Picard rank one by proving that tilt stable objects satisfy the strong Bogomolov-Gieseker type inequality. This is done by showing any Fourier-Mukai transform gives an equivalence of abelian categories which are double tilts of coherent sheaves.

The Euclid-Fourier-Mukai algorithm for elliptic surfaces

We describe the birational correspondences, induced by the Fourier-Mukai functor, between moduli spaces of semistable sheaves on elliptic surfaces with sections, using the notion of $P$-stability in the derived category. We give explicit conditions to determine whether these correspondences are isomorphisms. This is indeed not true in general and we describe the cases where the birational maps are Mukai flops. Moreover, this construction provides examples of new compactifications of the moduli spaces of vector bundles via sheaves with torsion and via complexes. We finally get for any fixed dim…

Fiberwise stable bundles on elliptic threefolds with relative Picard number one

We show that fiberwise stable vector bundles are preserved by relative Fourier-Mukai transforms between elliptic threefolds with relative Picard number one. Using these bundles we define new invariants of elliptic fibrations, and we relate the invariants of a space with those of a relative moduli space of stable sheaves on it. As a byproduct, we calculate the intersection form of a certain new example of an elliptic Calabi-Yau threefold.

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 Transforms and Stability Conditions on Abelian Varieties

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 formulate a conjecture which says certain Fourier-Mukai transforms between derived categories give equivalences of some hearts of Bridgeland stability conditions. We establish it for abelian surfaces, which is already known due to D. Huybrechts, and for abelian 3-folds. This generalizes the author's previous joint work with A. Maciocia on principally polarized abel…

Fourier-Mukai transforms of slope stable torsion-free sheaves on a product elliptic threefold

On the product elliptic threefold $X = C \times S$ where $C$ is an elliptic curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai transform $Φ$ on $D^b(X)$ induced by the classical Fourier-Mukai transform on $D^b(C)$, a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is taken to a limit tilt stable object. We also show that a limit tilt semistable object on $X$ is taken by $Φ$ to a slope semistable sheaf, up to mod…

Derived equivalences of twisted supersingular K3 surfaces

We study the derived categories of twisted supersingular K3 surfaces. We prove a derived crystalline Torelli theorem for twisted supersingular K3 surfaces, characterizing Fourier-Mukai equivalences in terms of isomorphisms between their associated K3 crystals. This is a positive characteristic analog of the Hodge-theoretic derived Torelli theorem of Orlov, and its extension to twisted K3 surfaces by Huybrechts and Stellari. We give applications to various questions concerning Fourier-Mukai partners, extending results of Căldăraru and Huybrechts and Stellari. We also give an exact formula for t…

Mukai's program for non-primitive curves on K3 surfaces

Mukai's program seeks to recover a K3 surface $X$ from any curve $C$ on it by exhibiting it as a Fourier-Mukai partner to a Brill-Noether locus of vector bundles on the curve. In the case $X$ has Picard number one and the curve $C\in |H|$ is primitive, this was confirmed by Feyzbakhsh for $g\geq 11$ and $g\neq 12$. More recently, Feyzbakhsh has shown that certain moduli spaces of stable bundles on $X$ are isomorphic to the Brill-Noether locus of curves in $|H|$ if $g$ is sufficiently large. In this paper, we work with irreducible curves in a non-primitive ample linear system $|mH|$ and prove t…

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…

On Derived Equivalences of K3 Surfaces in Positive Characteristic

For an ordinary K3 surface over an algebraically closed field of positive characteristic we show that every automorphism lifts to characteristic zero. Moreover, we show that the Fourier-Mukai partners of an ordinary K3 surface are in one-to-one correspondence with the Fourier-Mukai partners of the geometric generic fiber of its canonical lift. We also prove that the explicit counting formula for Fourier-Mukai partners of the K3 surfaces with Picard rank two and with discriminant equal to minus of a prime number, in terms of the class number of the prime, holds over a field of positive characte…

GV-sheaves, Fourier-Mukai transform, and Generic Vanishing

We use homological methods to establish a formal criterion for Generic Vanishing, in the sense originated by Green and Lazarsfeld and pursued further by Hacon and the first author, but in the context of an arbitrary Fourier-Mukai correspondence. For smooth projective varieties we apply this to deduce a Kodaira-type generic vanishing theorem for adjoint bundles of the form $K_X + L$ with $L$ a nef line bundle, and in fact a more general generic Nadel-type vanishing theorem for multiplier ideal sheaves. Still in the context of the Picard variety, the same method generates various other generic v…

The cohomology of the general stable sheaf on a K3 surface

Let $X$ be a K3 surface with Picard group $\mathrm{Pic}(X)\cong\mathbb{Z} H$ such that $H^2=2n$. Let $M_{H}(\mathbf{v})$ be the moduli space of Gieseker semistable sheaves on $X$ with Mukai vector $\mathbf{v}$. We say that $\mathbf{v}$ satisfies weak Brill-Noether if the general sheaf in $M_{H}(\mathbf{v})$ has at most one nonzero cohomology group. We show that given any rank $r \geq 2$, there are only finitely many Mukai vectors of rank $r$ on K3 surfaces of Picard rank one where weak Brill-Noether fails. We give an algorithm for finding the potential counterexamples and classify all such cou…