Ruan's Conjecture on Singular symplectic flops

Mirror of Atiyah flop in symplectic geometry and stability conditions

We study the mirror operation of the Atiyah flop in symplectic geometry. We formulate the operation for a symplectic manifold with a Lagrangian fibration. Furthermore we construct geometric stability conditions on the derived Fukaya category of the deformed conifold and study the action of the mirror Atiyah flop on these stability conditions.

The Berglund-Hübsch-Chiodo-Ruan mirror symmetry for K3 surfaces

We prove that the mirror symmetry of Berglund-Hübsch-Chiodo-Ruan, applied to K3 surfaces with a non-symplectic involution, coincides with the mirror symmetry described by Dolgachev and Voisin.

Q-Gorenstein环面接触流形的接触不变量、Ehrhart多项式与Chen-Ruan上同调

Contact invariants of Q-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

本文研究Q-Gorenstein环面接触流形的柱形接触同调不变量，建立其与环面图的Ehrhart（拟）多项式、对应环面孤立奇点的crepant环面轨形消解的Chen-Ruan上同调，以及生成该流形的单调环面辛轨形底空间的Chen-Ruan上同调的关联。

Mukai flops and deformations of symplectic resolutions

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.

Orbifold cohomology of abelian symplectic reductions and the case of weighted projective spaces

These notes accompany a lecture about the topology of symplectic (and other) quotients. The aim is two-fold: first to advertise the ease of computation in the symplectic category; and second to give an account of some new computations for weighted projective spaces. We start with a brief exposition of how orbifolds arise in the symplectic category, and discuss the techniques used to understand their topology. We then show how these results can be used to compute the Chen-Ruan orbifold cohomology ring of abelian symplectic reductions. We conclude by comparing the several rings associated to a w…

Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces

We study the birational geometry of irreducible holomorphic symplectic varieties arising as varieties of lines of general cubic fourfolds containing a cubic scroll. We compute the ample and moving cones, and exhibit a birational automorphism of infinite order explaining the chamber decomposition of the moving cone.

Transformation of algebraic Gromov-Witten invariants of three-folds under flops and small extremal transitions, with an appendix from the stringy and the symplectic viewpoint

We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic relative Gromov-Witten theory and its applications. This gives an algebro-geometric account of Witten's wall-crossing formula for correlation functions of the descendant nonlinear sigma model in adjacent geometric phases of a gauge linear sigma model and of the symplectic approach in an earlier work of An-Min Li and Yongbin Ruan on the same problem. A terse account from the stringy and the symplecti…

Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations

We compute the recently introduced Fan-Jarvis-Ruan-Witten theory of W-curves in genus zero for quintic polynomials in five variables and we show that it matches the Gromov-Witten genus-zero theory of the quintic three-fold via a symplectic transformation. More specifically, we show that the J-function encoding the Fan-Jarvis-Ruan-Witten theory on the A-side equals via a mirror map the I-function embodying the period integrals at the Gepner point on the B-side. This identification inscribes the physical Landau-Ginzburg/Calabi-Yau correspondence within the enumerative geometry of moduli of curve…

Degeneration and gluing of Kuranishi structures in Gromov-Witten theory and the degeneration/gluing axioms for open Gromov-Witten invariants under a symplectic cut

We study the degeneration and the gluing of Kuranishi structures in Gromov-Witten theory under a symplectic cut. This leads us to a degeneration axiom and a gluing axiom for open Gromov-Witten invariants. They provide then a route to the construction of virtual fundamental chains via specialization. Comments on the equivalence of the degeneration formula of closed Gromov-Witten invariants by Li-Ruan/Li versus Ionel-Parker are given in the Appendix.

Symplectic invariance of rational surfaces on Kähler manifolds

Kollar and Ruan proved symplectic deformation invariance for uniruledness of Kaehler manifolds. Zhiyu Tian proved the same for rational connectedness in dimension < 4. Kollar conjectured this in all dimensions. We prove Kollar's conjecture, as well as existence of a covering family of rational surfaces, for all Kaehler manifolds that are symplectically deformation equivalent to G/P or to a low degree complete intersection in such.

正辛流形中具有固定复结构的伪全纯曲线

Pseudo-holomorphic curves with a fixed complex structure in positive symplectic manifolds

本研究证明正辛流形中高次固定定义域Gromov-Witten不变量为伪全纯曲线的带符号计数，验证了Lian-Pandharipande猜想的辛版本成立，同时正面回应了Ruan与Tian的相关问题。

Wall-Crossings in Toric Gromov-Witten Theory II: Local Examples

In this paper we analyze six examples of birational transformations between toric orbifolds: three crepant resolutions, two crepant partial resolutions, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appropria…

Flops and Poisson deformations of symplectic varieties

This is a local version of math.AG/0506534. We shall deal with the deformation of a convex symplectic variety $X$ instead of a projective one. The usual deformation does not work well in the convex case. Instead, we regard $X$ as a Poisson scheme and study its Poisson deformation. One of the application is the following: Let $Y$ be an affine symplectic variety, and assume that $Y$ has two $Q$-factorial crepant terminalizations $X$ and $X'$. If $X$ is non-singular, then $X'$ is non-singular, too. Moreover, when $Y$ has a good $C^*$-action, $X$ and $X'$ have the same kind of singularities.

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…

Double bubble plumbings and two-curve flops

We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-dimensional spheres along a circle. These spaces are related, at a derived level and working in a characteristic determined by the specific geometry, to local threefolds which contain two floppable $(-1,-1)$-curves meeting at a point. Using contraction algebras we classify spherical objects on the B-side, and derive topological consequences including a complete description of the homology classes realised by graded exact Lagrangians.

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 …

On the Crepant Resolution Conjecture in the Local Case

In this paper we analyze four examples of birational transformations between local Calabi-Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov-Witten invariants, proving the Coates-Corti-Iritani-Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan's original Crepant Resolution Conjecture should be modified, by including appr…

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。

Topological properties of punctual Hilbert schemes of almost-complex fourfolds (II)

In this article, we study the rational cohomology rings of Voisin's punctual Hilbert schemes $X^{[n]}$ associated to a symplectic compact fourfold $X$. We prove that these rings can be universally constructed from $H^*(X,\mathbb{Q})$ and $c_1(X)$, and that Ruan's crepant resolution conjecture holds if $c_1(X)$ is a torsion class. Next, we prove that for any almost-complex compact fourfold $X$, the complex cobordism class of $X^{[n]}$ depends only on the cobordism class of $X$.

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…

Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I

We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete formula for the change of GW-invariants of any genus transform under flop and a general type I extremal transition. Other extremal transition will be handled in a subsequent paper.

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.

关于辛流形的泊松变形注释

Notes on Poisson deformations of symplectic varieties

本文对先前关于辛流形泊松变形的研究进行了修正和补充。

On the Chow ring of birational irreducible symplectic varieties

We show that the graded Chow rings of two birational irreducible symplectic varieties are isomorphic. This lifts a result known for the cohomology algebras to the level of Chow rings, despite the non-injectivity the cycle class map. In the special case of general Mukai flops, we present an alternative approach based on explicit calculations.

Duality theorems for étale gerbes on orbifolds

Let $G$ be a finite group and $\Y$ a $G$-gerbe over an orbifold $\B$. A disconnected orbifold $\hat{\Y}$ and a flat U(1)-gerbe $c$ on $\hat{\Y}$ is canonically constructed from $\Y$. Motivated by a proposal in physics, we study a mathematical duality between the geometry of the $G$-gerbe $\Y$ and the geometry of $\hat{\Y}$ {\em twisted by} $c$. We prove several results verifying this duality in the contexts of noncommutative geometry and symplectic topology. In particular, we prove that the category of sheaves on $\Y$ is equivalent to the category of $c$-twisted sheaves on $\hat{\Y}$. When $\Y…

On deformations of Q-factorial symplectic varieties

We shall prove that any small deformation of a Q-factorial projective symplectic variety with terminal singularities is locally rigid; in other words, it preserves the singularity. In particular, many singular symplectic moduli of semi-stable sheaves on K3 have no smoothings via deformations. As an application of the result, we also prove that the smootheness is preserved under a flop in our symplectic case. We conjecture that a projective symplectic variety has a smoothing by a flat deformation if and only if it has a crepant (symplectic) resolution. This conjecture would be true if the minim…

ADE-轨形等变上同调创消解猜想

Equivariant Cohomological Crepant Resolution Conjecture for ADE-orbifolds

本文证明了ADE轨形[C^2/G]的C^*-等变上同调创消解猜想，通过McKay对应和特征标理论，将Bryan-Gholampour变换化为统一根系恒等式，并给出等变Chen-Ruan乘积的显式表示。

Motivic HyperKähler Resolution Conjecture : I. Generalized Kummer varieties

Given a smooth projective variety $M$ endowed with a faithful action of a finite group $G$, following Jarvis-Kaufmann-Kimura and Fantechi-Göttsche, we define the orbifold motive (or Chen-Ruan motive) of the quotient stack $[M/G]$ as an algebra object in the category of Chow motives. Inspired by Ruan, one can formulate a motivic version of his Cohomological HyperKähler Resolution Conjecture. We prove this motivic version, as well as its K-theoretic analogue conjectured by Jarvis-Kaufmann-Kimura, in two situations related to an abelian surface $A$ and a positive integer $n$. Case (A) concerns Hi…

The Hochschild cohomology ring of a global quotient orbifold

We study the cup product on the Hochschild cohomology of the stack quotient [X/G] of a smooth quasi-projective variety X by a finite group G. More specifically, we construct a G-equivariant sheaf of graded algebras on X whose G-invariant global sections recover the associated graded algebra of the Hochschild cohomology of [X/G], under a natural filtration. This sheaf is an algebra over the polyvector fields T^{poly}_X on X, and is generated as a T^{poly}_X-algebra by the sum of the determinants det(N_{X^g}) of the normal bundles of the fixed loci in X. We employ our understanding of Hochschild…

Birational Geometry of Singular Moduli Spaces of O'Grady Type

Following Bayer and Macrì, we study the birational geometry of singular moduli spaces $M$ of sheaves on a K3 surface $X$ which admit symplectic resolutions. More precisely, we use the Bayer-Macrì map from the space of Bridgeland stability conditions $\mathrm{Stab}(X)$ to the cone of movable divisors on $M$ to relate wall-crossing in $\mathrm{Stab}(X)$ to birational transformations of $M$. We give a complete classification of walls in $\mathrm{Stab}(X)$ and show that every birational model of $M$ obtained by performing a finite sequence of flops from $M$ appears as a moduli space of Bridgeland …