Derived equivalence for stratified Mukai flop on G(2,4)

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

Mukai flops and Plücker type formulas for hyper-Kähler manifolds

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.

Mukai flops and P-twists

Associated to a Mukai flop X ---> X' is on the one hand a sequence of equivalences D(X) -> D(X'), due to Kawamata and Namikawa, and on the other hand a sequence of autoequivalences of D(X), due to Huybrechts and Thomas. We work out a complete picture of the relationship between the two. We do the same for standard flops, relating Bondal and Orlov's derived equivalences to spherical twists, extending a well-known story for the Atiyah flop to higher dimensions.

Mukai flops and derived categories

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…

Grassmann 翻折的墙跨越与 Fourier-Mukai 变换

Wall Crossing and the Fourier-Mukai Transform for Grassmann Flops

本文证明光滑基上相对 Grassmann 翻折的 crepant 变换猜想，通过解析延拓与辛变换联系相应 GIT 商空间的 $I$-函数，并验证该辛变换与 Iritani 积分结构相容、由 $K$-理论中的 Fourier-Mukai 变换诱导。

On Mukai flops for Scorza varieties

I give three descriptions of the Mukai flop of type $E\_{6,I}$, one in terms of Jordan algebras, one in terms of projective geometry over the octonions, and one in terms of O-blow-ups. Each description shows that it is very similar to certain flops of type $A$. The Mukai flop of type $E\_{6,II}$ is also described.

Birational Geometry of symplectic resolutions of nilpotent orbits

In this paper, we shall prove that any two (projective) symplectic resolutions of a nilpotent orbit closure in a classical simple Lie algebra are connected by a finite sequence of diagrams which are locally trivial families of Mukai flops of type A or of type D. A Mukai flop of type A is the same as a stratified Mukai flop in our previous paper "Mukai flops and derived categories II". On the other hand, a Mukai flop of type D comes from a nilpotent orbit in so(2k). Let G^+ and G^- be two connected components of the orthogonal Grassmannian G_{iso}(k,2k). Then the closure of this orbit has two S…

On certain K-equivalent birational maps

We study K-equivalent birational maps which are resolved by a single blowup. Examples of such maps include standard flops and twisted Mukai flops. We give a criterion for such maps to be a standard flop or a twisted Mukai flop. As an application, we classify all such birational maps up to dimension 5.

Mukai Flop and Ruan Cohomology

Suppose that two compact manifolds $X, X'$ are connected by a sequence of Mukai flops. In this paper, we construct a ring isomorphism between cohomology ring of $X$ and $X'$. Using the local mirror symmetry technique, we prove that the quantum corrected products on $X, X'$ are the ordinary intersection products. Furthermore, $X, X'$ have isomorphic Ruan cohomology. i.e. we proved the cohomological minimal model conjecture proposed by Ruan.

Equivalences and stratified flops

We construct natural equivalences between derived categories of coherent sheaves on the local models for stratified Mukai or Atiyah flops (of type A).

Flops and derived categories

This paper contains some applications of Fourier-Mukai techniques to the birational geometry of threefolds. In particular, we prove that birational Calabi-Yau threefolds have equivalent derived categories. To do this we show how flops arise naturally as moduli spaces of perverse coherent sheaves.

Flops and Equivalences of derived Categories for Threefolds with only Gorenstein Singularities

The main propose of this paper is to show that Bridgeland's moduli space of perverse point sheaves for certain flopping contractions gives the flops, and the Fourier-Mukai transform given by the birational correspondence of the flop is an equivalence between bounded derived categories.

Birational geometry of symplectic resolutions of nilpotent orbits II

In this paper we shall study symplectic resolutions of a nilpotent orbit closure of a complex simple Lie algebra \g. We shall introduce an equivalence relation in the set of parabolic subgroups of $G$ in terms of marked Dynkin diagrams. We start with a nilpotent orbit closure which admits a Springer resolution with a parabolic subgroup $P_0$ of $G$. Then we prove that all symplectic resolution of the nilpotent closure are Springer resolutions with $P$ which are equivalent to $P_0$. Here all symplectic resolutions are connected by Mukai flops. We need three types of Mukai flops (types A, D and …

超凯勒流形的双有理几何与Hu-Yau猜想

Birational geometry of hyperkahler manifolds and the Hu-Yau conjecture

针对超凯勒流形双亚纯映射分解的Hu-Yau猜想，研究者通过将映射分解为与MBM收缩相关的墙跨越翻转变换复合，证明了该猜想在最大完整和乐群的紧超凯勒流形情形下成立。

Brill-Noether duality for moduli spaces of sheaves on K3 surfaces

Components of the Moduli space of sheaves on a K3 surface are parametrized by a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often lift to symplectic birational isomorphisms of the collection of components. An example of such a birational isomorphism is the Abel-Jacobi map relating the Hilbert scheme of g points on a K3 of degree 2g-2 to an integrable system: the union of Jacobians of hyperplane sections (curves) of genus g. The main results are: 1) We construct a stratified version of a Mukai elementary transformation modeled after dual pairs of Springer resolutions…

Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops

In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure $\overline{B(1)}$ of type A, and study relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$ as moduli spaces of representations of the quiver. We also study the Kawamata-Namikawa's derived equivalence between crepant resolutions $Y$ and $Y^+$ of $\ove…

Spherical functors and the flop-flop autoequivalence

Flops are birational transformations which, conjecturally, induce derived equivalences. In many cases an equivalence can be produced as pull-push via a resolution of the birational transformation; when this happens, we have a non-trivial autoequivalence of either sides of the flop known as the \emph{flop-flop autoequivalence}. We prove that such autoequivalence can be realised as the inverse of a spherical twist around a conservative, spherical functor in a natural way. More precisely, we prove that a natural, conservative spherical functor exists in a more general framework and that the flop-…

Flops and about: a guide

Stratified flops show up in the birational geometry of symplectic varieties such as resolutions of nilpotent orbits and moduli spaces of sheaves. Constructing derived equivalences between varieties related by such flops is, strangely enough, related to areas in representation theory and knot homology. In this paper we discuss how to construct such equivalences, explain the main tool for doing this (categorical Lie algebra actions) and comment on various related topics.

Windows for cdgas

We study a Fourier-Mukai kernel associated to a GIT wall-crossing for arbitrarily singular (not necessarily reduced or irreducible) affine varieties over any field. This kernel is closely related to a derived fiber product diagram for the wall-crossing and simple to understand from the viewpoint of commutative differential graded algebras. However, from the perspective of algebraic varieties, the kernel can be quite complicated, corresponding to a complex with multiple homology sheaves. Under mild assumptions in the Calabi-Yau case, we prove that this kernel provides an equivalence between the…

Deformation of tilting-type derived equivalences for crepant resolutions

We say that an exact equivalence between the derived categories of two algebraic varieties is tilting-type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.

Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient

For a finite subgroup G in SL(3,C), Bridgeland, King and Reid proved that the moduli space of G-clusters is a crepant resolution of the quotient C^3/G. This paper considers the moduli spaces M_θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of the GIT parameter θ. For G Abelian, we prove that every projective crepant resolution of C^3/G is isomorphic to M_θfor some parameter θ. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier--Mukai transform. We also unc…

通过范畴 sl(2) 作用的窗口等价

Window equivalences via categorical sl(2) actions

本文通过 Knörrer 周期性，统一了格拉斯曼余切丛分层 Mukai 翻转的两种导出等价方法，即几何范畴 sl(2) 作用与魔窗范畴。

Moduli spaces of torsion sheaves on K3 surfaces and derived equivalences

We show that for many moduli spaces M of torsion sheaves on K3 surfaces S, the functor D(S) -> D(M) induced by the universal sheaf is a P-functor, hence can be used to construct an autoequivalence of D(M), and that this autoequivalence can be factored into geometrically meaningful equivalences associated to abelian fibrations and Mukai flops. Along the way we produce a derived equivalence between two compact hyperkaehler 2g-folds that are not birational, for every g >= 2. We also speculate about an approach to showing that birational moduli spaces of sheaves on K3 surfaces are derived-equivale…