Misha Verbitsky
Paper withdrawn due to errors (superseded by math.AG/0604303). Proposition 11.4 is false, Section 12 is false, and the main statement is true only for bundles $B$ with $c_1(B)$ SU(2)-invariant.
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Any minimal model of a projective Hyperkaehler manifold is a projective Hyperkaehler manifold. As a consequence, moduli spaces of sheaves on a k3 that don't admit a symplectic resolution are not birational to Hyperkaehler manifolds.
We study the algebraic dimension a(X) of a compact hyperkaehler manfold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact Kaehler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0,n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If n = 2, then - without any assumptions - the algebraic dimension only takes the values 0,2 and 4. The paper gives structure results for "generalised hyperkaehler" manifolds and studies nef lines bundles.
We address the following question: what are the cardinalities of maximal finite families of pairwise incident planes in a complex projective space? One proves easily that the span of the planes has dimension 5 or 6. Up to projectivities there is one such family spanning a 6-dimensional projective space - this is an elementary result. Maximal finite families of pairwise incident planes in a 5-dimensional projective space are considerably more misterious: they are linked to certain special (EPW) sextic hypersurfaces which have a non-trivial double cover, generically a hyperkaehler 4-fold. We pro…
In this paper we classify codimension 1 Mukai foliations on complex projective manifolds
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.
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 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.
We investigate a construction providing pairs of Calabi-Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described by Kanemitsu. We discuss the implications of such construction at the level of Hodge equivalence, derived equivalence and $\mathbb L$-equivalence. For the case of $K3$ surfaces, we provide alternative interpretations for the Fourier-Mukai duality in the family of $K3$ surfaces of degree 12 of Mukai. In all these constructions the derived equivalence lifts to an equivalence of matrix factorizations categories.
We shall show that the Picard number of the generic fiber, in the sense of scheme, of an abelian fibered hyperkähler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O'Grady's 10-dimensional manifold, we construct an abelian fibered hyperkähler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.
We prove that two projective symplectic resolutions of $\cit^{2n}/G$ are connected by Mukai flops in codimension 2 for a finite sub-group $G < \Sp(2n)$. It is also shown that two projective symplectic resolutions of $\cit^4/G$ are deformation equivalent.
Our aim in this article is to produce new examples of semistable Lazarsfeld- Mukai bundles on smooth projective surfaces $X$ using the notion of parabolic vector bundles. In particular, we associate natural parabolic structures to any rank two (dual) Lazarsfeld-Mukai bundle and study the parabolic stability of these parabolic bundles. We also show that the orbifold bundles on Kawamata coverings of $X$ corresponding to the above parabolic bundles are themselves certain (dual) Lazarsfeld-Mukai bundles. This gives semistable Lazarsfeld-Mukai bundles on Kawamata covers of the projective plane and …
In this paper we prove that any smooth projective variety of dimension $\ge 3$ equipped with a tilting bundle can serve as the source variety of a non-Fourier-Mukai functor between smooth projective schemes.
This article contains a compression of results from alg-geom/9501001, with most proofs omitted. We prove that every two points of the connected moduli space of holomorphically symplectic manifolds can be connected with so-called ``twistor lines'' -- projective lines holomorphically embedded to the moduli space and corresponding to the hyperkähler structures. This has interesting implications for the geometry of compact hyperkähler manifolds and of holomorphic vector bundles over such manifolds.
We prove that the characteristic foliation $F$ on a non-singular divisor $D$ in an irreducible projective hyperkaehler manifold $X$ cannot be algebraic, unless the leaves of $F$ are rational curves or $X$ is a surface. More generally, we show that if $X$ is an arbitrary projective manifold carrying a holomorphic symplectic $2$-form, and $D$ and $F$ are as above, then $F$ can be algebraic with non-rational leaves only when, up to a finite étale cover, $X$ is the product of a symplectic projective manifold $Y$ with a symplectic surface and $D$ is the pull-back of a curve on this surface. When $D…
We study the intersection theory of complex Lagrangian subvarieties inside holomorphic symplectic manifolds. In particular, we study their behaviour under Mukai flops and give a rigorous proof of the Plücker type formula for Legendre dual subvarieties written down by the second author before. Then we apply the formula to study projective dual varieties in projective spaces.
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。
Due to a theorem by Orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of Fourier-Mukai type. We extend this result to the case of bounded derived categories of twisted coherent sheaves and at the same time we weaken the hypotheses on the functor. As an application we get a complete description of the exact functors between the abelian categories of twisted coherent sheaves on smooth projective varieties.
Let X be a hyperkaehler manifold of Kummer type. We study the 4 dimensional intermediate Jacobian J^3(X) constructed out of the 3rd cohomology of X, motivated by the desire to understand the Kuga-Satake torus associated to X. We prove that there is an embedding of Hodge structures of H^2(X) into H^2(J^3(X)). It follows that if X is projective with polarization L, the projective Kuga-Satake torus of (X,L) is isogenous to the fourth power of J^3(X). By studying the cohomology ring of X, we find that there is a natural smooth quadric Q(X) in the projectivization of H^3(X), with a natural choice o…
In this note, we shall prove that two smooth projective varieties of dim 2n connected by a Mukai flop have equivalent bounded derived categories. More precisely, let $ϕ: X - - \to X^+$ be a Mukai flop with centers $Y \subset X$ and $Y^+ \subset X^+$. In our case, the natural fuctor $Φ: D(X) \to D(X^+)$ defined by the graph of $ϕ$ is not fully faithful. Instead, let $X \to {\bar X}$ and $X^+ \to {\bar X}$ be the birational contraction maps of the centers, and put ${\hat X} := X \times_{{\bar X} X^+$. Then ${\hat X}$ is a normal crossing variety with two irreducible components. This ${\hat X}$ d…
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 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.
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 $κ(X)=dim X-1$.
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.
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.
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.
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.
We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic equations on a triple -- vortices -- and explore some of its basic properties. Our approach combines direct methods with dimensional reduction techniques, relating triples over a curve with vector bundles over the product of the curve with the complex projective line.
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.
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 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.
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.
On Mukai's conjecture for hyperelliptic varieties
本文研究数值平凡典范线丛的光滑射影簇的合冲性质，证明了一般性结果，并在任意维数和正特征下验证了阿贝尔簇有限商的Mukai合冲猜想若干情形。
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.
Fourier--Mukai transforms commuting with Frobenius
本文证明了特征 $p$ 光滑射影簇间与 Frobenius 前推或拉回交换的 Fourier--Mukai 等价，可分解为平移、同构及与特定可逆层张量积的复合。
Let $X$ be a smooth projective K3 surface over complex numbers and $C$ be an ample curve on $X$. In this paper we will study the semistability of the Lazarsfeld-Mukai bundle $E_{C, A}$ associated to a line bundle $A$ ion $C$ such that $|A|$ is a pencil on $C$ and computes the Clifford index of $C$. We give a necessary and sufficient condition for $E_{C, A}$ being semistable.
We prove that exact functors between the categories of perfect complexes supported on projective schemes are of Fourier--Mukai type if the functor satisfies a condition weaker than being fully faithful. We also get generalizations of the results in the literature in the case without support conditions. Some applications are discussed and, along the way, we prove that the category of perfect supported complexes has a strongly unique enhancement.
We discuss the following problem: how can an arbitrary Fourier-Mukai transform $ϕ: \mathrm{D}^{\mathrm{b}}( \mathbb{P}^a ) \rightarrow \mathrm{D}^{\mathrm{b}}( \mathbb{P}^b )$ between the bounded derived categories of two projective spaces of dimensions $a$ and $b$ be expressed in explicit terms as an exact functor between the homotopy categories $\mathrm{K}^{\mathrm{b}}( \mathbb{B}^a ) \rightarrow \mathrm{K}^{\mathrm{b}}( \mathbb{B}^b )$ generated by the full strong exceptional sequences of the line bundles $\mathbb{B}^a = \{\mathcal{O}(-a), \dots, \mathcal{O}\}$ and $\mathbb{B}^b = \{\mathca…
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.
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 …
Consider an ample and globally generated line bundle $L$ on a smooth projective variety $X$ of dimension $N\geq 2$ over $\mathbb{C}$. Let $D$ be a smooth divisor in the complete linear system of $L$. We construct reflexive sheaves on $X$ by an elementary transformation of a trivial bundle on $X$ along certain globally generated torsion-free sheaves on $D$. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the $μ_L$-(semi)stability of such reflexive sheaves under certain conditions.
Orlov's famous representability theorem asserts that any fully faithful functor between the derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. This result has been extended by Lunts and Orlov to include functors from perfect complexes to quasi-coherent complexes. In this paper we show that the latter extension is false without the full faithfulness hypothesis. Our results are based on the properties of scalar extensions of derived categories, whose investigation was started by Pawel Sosna and the first author.
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 …
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。
A normal projective variety X is called Fano if a multiple of the anticanonical Weil divisor, -K_X, is an ample Cartier divisor, the index of a Fano variety is the number i(X):=sup{t: -K_X= tH, for some ample Cartier divisor H}. Mukai announced, the classification of smooth Fano manifolds X of index i(X)=n-2, under the assumption that the linear system |H| contains a smooth divisor. In this paper we prove that this assumption is always satisfied. Therefore the result of Mukai provide a complete classification of smooth Fano n-folds of index $i(X)=n-2$, Mukai manifolds.
Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$. For a vector bundle $E$ on $S$, let ${\mathcal H}(E)\, \longrightarrow\, {\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of $S\times {\rm Hilb}^d(S)$ as the kernel. We prove that two vector bundles $E$ and $F$ on $S$ are isomorphic if the vector bundles ${\mathcal H}(E)$ and ${\mathcal H}(F)$ …
On Fourier-Mukai transforms of upward flows for Hitchin systems
本文利用Arinkin层和余维数估计，将Hausel和Hitchin关于Donagi-Pantev对偶在光滑Hitchin纤维开集上交织拉格朗日向上流与超全纯向量丛的结果，推广至整个Hitchin底上任意次数的Higgs丛。
To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) Calabi-Yau projective varieties with locally complete intersection singularities. We define a Poincare' sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the integral transform with kernel the Poincare' sheaf is an equivalence of their derived categories. In particular, any two fine compactified Jacobians are derived equivalent. When applied…
We study moduli spaces of sheaves over non-projective K3 surfaces. More precisely, if $v=(r,ξ,a)$ is a Mukai vector on a K3 surface $S$ with $r$ prime to $ξ$ and $ω$ is a "generic" Kähler class on $S$, we show that the moduli space $M$ of $μ_ω-$stable sheaves on $S$ with associated Mukai vector $v$ is an irreducible holomorphic symplectic manifold which is deformation equivalent to a Hilbert scheme of points on a K3 surface. If $M$ parametrizes only locally free sheaves, it is moreover hyperkähler. Finally, we show that there is an isometry between $v^{\perp}$ and $H^{2}(M,\mathbb{Z})$ and tha…
In the first part of this paper we prove a vanishing criterion for higher direct images of projective families of line bundles on a Cohen-Macaulay variety X. The result involves certain first-order deformations of certain curves on X, and makes essential use of the notion of global co-gaussian maps, a generalization of Wahl's gaussian maps. In the second part we apply the criterion above, combined with Fourier-Mukai transform on abelian varieties, to prove an algebraic version of Green-Lazarsfeld's Generic Vanishing Theorem. In fact we prove a stronger result concerning higher direct images of…
We extend to singular schemes with Gorenstein singularities or fibered in schemes of that kind Bondal and Orlov's criterion for an integral functor to be fully faithful. We also contemplate a criterion for equivalence. We offer a proof that is new even if we restrict to the smooth case. In addition, we prove that for locally projective Gorenstein morphisms, a relative integral functor is fully faithful if and only if its restriction to each fibre also is it. These results imply the invertibility of the usual relative Fourier-Mukai transform for an elliptic fibration as a direct corollary.
In a suitable regime of superstring theory, D-branes in a Calabi-Yau space and their most fundamental behaviors can be nicely described mathematically through morphisms from Azumaya spaces with a fundamental module to that Calabi-Yau space. In the earlier work [L-L-S-Y] (D(2): arXiv:0809.2121 [math.AG], with Si Li and Ruifang Song) from the project, we explored this notion for the case of D1-branes (i.e. D-strings) and laid down some basic ingredients toward understanding the notion of D-string world-sheet instantons in this context. In this continuation, D(10), of D(2), we move on to construc…
Inspired by their results on the Chow rings of projective K3 surfaces, Beauville and Voisin made the following conjecture: given a projective hyperkaehler manifold, for any algebraic cycle which is a polynomial with rational coefficients of Chern classes of the tangent bundle and line bundles, it is rationally equivalent to zero if and only if it is numerically equivalent to zero. In this paper, we prove the Beauville-Voisin conjecture for generalized Kummer varieties.
Twisted Hodge diamonds give rise to non-Fourier-Mukai functors
本文利用扭曲霍奇菱形计算霍赫希尔德维数，构造并验证了奇数维光滑二次超曲面上大量非傅里叶-向井函子，并证明每个奇数维光滑二次超曲面至少存在一个这样的函子。
Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tauto…
In this paper we study the second integral cohomology of moduli spaces of semistable sheaves on projective K3 surfaces. If $S$ is a projective K3 surface, $v$ a Mukai vector and $H$ a $v-$generic polarization on $S$, we show that $H^{2}(M_{v},\mathbb{Z})$ is a free $\mathbb{Z}-$module of rank 23 carrying a pure weight-two Hodge structure and a lattice structure, with respect to which $H^{2}(M_{v},\mathbb{Z})$ is Hodge isometric to the Hodge sublattice $v^{\perp}$ of the Mukai lattice of $S$. Similar results are proved for Abelian surfaces.
Curves on Brill-Noether special K3 surfaces
本文利用 Lazarsfeld-Mukai 向量丛技巧，对亏格不超过 19 的极化 K3 曲面证明了其 Brill-Noether 一般性等价于线性系中光滑曲线的 Brill-Noether 一般性，从而肯定回答了 Knutsen 与 Mukai 提出的公开问题。
Fourier-Mukai transform for fine compactified Prym varieties
本文针对由平面奇异谱曲线定义的紧化 Prym 簇构造 Fourier–Mukai 变换，建立其与等变导出范畴的等价，从而得到稠密一类奇异谱曲线上 SL(n,C) 与 PGL(n,C) Hitchin 纤维的导出等价。
Fiberwise criteria for Fourier--Mukai equivalences
本文研究积分变换在基变换下的行为，提出局部代数和纤维的瑜伽方法，推广Orlov的结果至奇异变分，并在算术设定中提供新的纤维和奇异性洞察。
Fourier-Mukai transforms and normalisation of nodal curves
本文研究了阿林金的庞加莱sheaf在紧化雅可比簇奇异点上的性质，通过正则化得到庞加莱sheaf之间的关系，利用抛物模的模空间进行sheaf数据的交织，为后续研究奇点处的希奇金系统镜像对称提供基础。
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