David Favero
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.
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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.
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.
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。
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.
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 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.
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.
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.
Arithmetic finiteness of Mukai varieties of genus 7
该研究针对亏格7的素Fano三维簇及高维Mukai簇的算术有限性展开，得到Torelli定理的算术精细化、Shafarevich型有限性的维数差异结果，并明确了定义在整数环上的这类簇的存在维数范围。
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.
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。
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。
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.
We shall give a complete geometrical description of the FM partners of a K3 surface of Picard number 1 and its applications.
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。
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-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。
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.
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。
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。
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。
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…
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。
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。
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.
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 shall give a Counting Formula for the number of Fourier-Mukai partners of a K3 surface and consider three applications.
Gluing of Fourier-Mukai partners in a triangular spectrum and birational geometry
本文在三角谱中构造Fourier-Mukai轨迹以粘合FM伙伴的张量三角谱，结合实例揭示其与代数簇几何、双有理性质的关联，并对比相关轨迹提出关于Serre不变轨迹的精确猜想。
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。
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.
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 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…
Derived equivalence of elliptic K3 surfaces and Jacobians
本文研究 K3 的 Fourier–Mukai 伙伴上的导出椭圆结构，利用 Hodge 数据与判别群完成分类，并据此回答椭圆 K3 曲面伙伴是否均为 Jacobian 的问题。
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…
We study generalized complex structures on K3 surfaces, in the sense of Hitchin. For each real parameter t between one and infinity we exhibit two families of generalized K3 surfaces, (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}), parametrized by zeta in CP^1, which are Mukai dual for zeta=0 and infinity, amd mirror partners for zeta not equal to 0 and infinity. Moreover, the Fourier-Mukai equivalence D^b(M,cal{I}_0) -> D^b(M,cal{J}_0) induces an isomorphism phi_T between the spaces of first order deformations of (M,cal{I}_0) and (M,cal{J}_0) as generalized complex manifolds, and the deformations (M…
In [Ma1] S. Ma established a bijection between Fourier--Mukai partners of a K3 surface and cusps of the Kähler moduli space. The Kähler moduli space can be described as a quotient of Bridgeland's stability manifold. We study the relation between stability conditions $σ$ near to a cusp and the associated Fourier--Mukai partner Y in the following ways. (1) We compare the heart of $σ$ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of $σ$-stable objects. An appendix is devoted to the group of auto-equivalences of the derived category which respect the component $Stab^\da…
A new proof of the Bondal-Orlov reconstruction using Matsui spectra
本文利用 Matsui 谱证明拟射影簇的完美导出范畴的谱嵌入关系，并据此给出 Bondal-Orlov 与 Ballard 重构定理的新证明，进而重构阿贝尔簇的 Fourier-Mukai 伙伴。
We prove that any Fourier--Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the Fourier--Mukai set of canonical covers of hyperelliptic and Enriques surfaces over an algebraically closed field of characteristic greater than three is trivial. These results extend to positive characteristic earlier results of Bridgeland--Maciocia and Sosna.
Algebraic cycles on Gushel-Mukai varieties
研究复 Gushel-Mukai (GM) 簇上的代数环，证明了广义 Hodge 猜想、(动机) Mumford-Tate 猜想和广义 Tate 猜想，计算了除两个无限维情形外的所有积分 Chow 群，并证明了广义伙伴或广义对偶的 GM 簇在中度有理 Chow 动机上同构。
We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures, posed by T. Shioda 25 years ago.
A rational Lagrangian fibration f on an irreducible symplecitc variety V is a rational map which is birationally equivalent to a regular surjective morphism with Lagrangian fibers. By analogy with K3 surfaces, it is natural to expect that a rational Lagrangian fibration exists if and only if V has a divisor D with Bogomolov--Beauville square 0. This conjecture is proved in the case when V is the punctual Hilbert scheme of a generic algebraic K3 surface S. The construction of f uses a twisted Fourier--Mukai transform which induces a birational isomorphism of V with a certain moduli space of twi…
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