Flops and Poisson deformations of symplectic varieties
Yoshinori Namikawa
备注：
In the previous version, we claimed that singularities do not change under an arbitrary symplectic flop. But, this claim is not justified for lack of algebraization
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.
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.
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.
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.
In this paper, we study the symplectic geometry of singular conifolds of the finite group quotient $$ W_r=\{(x,y,z,t)|xy-z^{2r}+t^2=0 \}/μ_r(a,-a,1,0), r\geq 1, $$ which we call orbi-conifolds. The related orbifold symplectic conifold transition and orbifold symplectic flops are constructed. Let $X$ and $Y$ be two symplectic orbifolds connected by such a flop. We study orbifold Gromov-Witten invariants of exceptional classes on $X$ and $Y$ and show that they have isomorphic Ruan cohomologies. Hence, we verify a conjecture of Ruan.
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…
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.
This paper deals with symplectic varieties which do not have symplectic resolutions. Some moduli spaces of semi-stable torsion-free sheaves on a K3 surface, and symplectic V-manifolds are such varieties. We shall prove local Torelli theorem for symplectic varieties. Some results on symplectic singularities are also included.
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 …
We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.
We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
We prove that under semismall smooth flops, smooth projective varieties have (non-canonically) isomorphic Chow motives with coefficients in any noetherian local ring $Λ$ by comparing the pushforward of the constant intersection complexes through flopping contractions.
We study derived equivalences of Abelian varieties in terms of their associated symplectic data. For simple Abelian varieties over an algebraically closed field of characteristic zero we prove that the natural correspondence introduced by Orlov, which maps equivalences to symplectic isomorphisms, is surjective.
We prove that the integral Hodge conjecture holds for 1-cycles on irreducible holomorphic symplectic varieties of K3 type and of Generalized Kummer type. As an application, we give a new proof of the integral Hodge conjecture for cubic fourfolds.
We study the global geometry of the ten dimensional O'Grady irreducible symplectic variety. We determine its second Betti number, its Beauville form and its Fujiki constant.
We prove that any ample class on a primitive symplectic variety that is locally trivial deformation of O'Grady's singular 6 dimensional example is proportional to the first Chern class of a uniruled divisor. This result answers a question of Lehn, Mongardi and Pacienza, extending their result for primitive symplectic varieties of this deformation type.
We study families of rational curves on certain irreducible holomorphic symplectic varieties. In particular, we prove that any ample linear system on a projective holomorphic symplectic variety of K3[n]-type contains a uniruled divisor. As an application we provide a generalization of the Beauville-Voisin result on the Chow group of 0-cycles on such varieties.
We give an example of a pair of projective symplectic varieties in arbitrarily large dimensions which are D-equivalent, L-equivalent, and birationally inequivalent.
We determine the Mori cone of holomorphic symplectic varieties deformation equivalent to the punctual Hilbert scheme on a K3 surface. Our description is given in terms of Markman's extended Hodge lattice.
We show that the Andre motive of an irreducible, symplectic variety, which is deformation equivalent to the Hilbert scheme of points on a K3 surface, is an object the category generated by its motive truncated in degree two.
Under some conditions on the deformation type, which we expect to be satisfied for arbitrary irreducible symplectic varieties, we describe which big and nef line bundles on irreducible symplectic varieties have base divisors. In particular, we show that such base divisors are always irreducible and reduced. This is applied to understand the behaviour of divisorial base components of big and nef line bundles under deformations and for K3$^{[n]}$-type and Kum$^n$-type.
We prove that the Kodaira dimension of the n-fold universal family of lattice-polarized holomorphic symplectic varieties with dominant and generically finite period map stabilizes to the moduli number when n is sufficiently large. Then we study the transition of Kodaira dimension explicitly, from negative to nonnegative, for known explicit families of polarized symplectic varieties. In particular, we determine the exact transition point in the cases of Beauville-Donagi and Debarre-Voisin, where the Borcherds Phi_{12} form plays a crucial role.
If X is a symplectic variety emedded in an affine space as a complete intersection of homogeneous polynomials, then X coincides with the nilpotent variety of a semisimple Lie algebra.
We give a new proof of the derived equivalence of a pair of varieties connected by the flop of type $C_2$ in the list of Kanemitsu arXiv:1812.05392, which is originally due to Segal. We also prove the derived equivalence of a pair of varieties connected by the flop of type $A^G_4$ in the same list. The latter proof follows that of the derived equivalence of Calabi--Yau 3-folds in Grassmannians $\text{Gr}(2,5)$ and $\text{Gr}(3,5)$ by Kapustka and Rampazzo arXiv:1711.10231 closely.
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.
A singular Moishezon space is not projective even if it is Kaehler. This paper gives a projectivity criterion for Kaehler Moishezon spaces. We shall use this projectivity criterion to show the density of projective symplectic varieties in the Kuranishi family of a (singular) symplectic variety.
Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.
We shall prove that the Poisson deformation functor of an affine (singular) symplectic variety is unobstructed. As a corollary, we prove the following result. Theorem: For an affine symplectic variety $X$ with a good $C^*$-action (where its natural Poisson structure is positively weighted), the following are equivalent (1) $X$ has a crepant projective resolution. (2) $X$ has a smoothing by a Poisson deformation.
We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …
We prove the standard conjectures for complex projective varieties that are deformations of the Hilbert scheme of points on a K3 surface. The proof involves Verbitsky's theory of hyperholomorphic sheaves and a study of the cohomology algebra of Hilbert schemes of K3 surfaces.
We show that the moduli spaces of sheaves on a projective K3 surface are irreducible symplectic varieties, and that the same holds for the fibers of the Albanese map of moduli spaces of sheaves on an Abelian surface.
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…
We propose a general framework governing the intersection properties of extremal rays of irreducible holomorphic symplectic manifolds under the Beauville-Bogomolov form. Our main thesis is that extremal rays associated to Lagrangian projective subspaces control the behavior of the cone of curves. We explore implications of this philosophy for examples like Hilbert schemes of points on K3 surfaces and generalized Kummer varieties. We also collect evidence supporting our conjectures in specific cases.
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…
We define the odd symplectic grassmannians and flag manifolds, which are smooth projective varieties equipped with an action of the odd symplectic group and generalizing the usual symplectic grassmannians and flag manifolds. Contrary to the latter, which are the flag manifolds of the symplectic group, the varieties we introduce are not homogeneous. We argue nevertheless that in many respects the odd symplectic grassmannians and flag manifolds behave like homogeneous varieties; in support of this claim, we compute the automorphism group of the odd symplectic grassmannians, and we prove a Borel-…
For any simple, simply connected algebraic group $G$ of type $B,C$ and $D$ and for any maximal parabolic subgroup $P$ of $G$, we provide a criterion for a Richardson variety in $G/P$ to admit semistable points for the action of a maximal torus $T$ with respect to an ample line bundle on $G/P$.
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.
We survey recent results about the Torelli question for holomorphic-symplectic varieties. Following are the main topics. A Hodge theoretic Torelli theorem. A study of the subgroup W, of the isometry group of the weight 2 Hodge structure, generated by reflection with respect to exceptional divisors. A description of the birational Kahler cone as a fundamental domain for the W-action on the positive cone. A proof of a weak version of Morrison's movable cone conjecture. A description of the moduli spaces of polarized holomorphic symplectic varieties as monodromy quotients of period domains of typ…
Let $\mbox{IG}(k,2n+1)$ be the odd symplectic Grassmannian. It is a quasi-ho\-mo\-ge\-neous space with homogeneous-like behavior. A very limited description of curve neighborhoods of Schubert varieties in $\mbox{IG}(k,2n+1)$ was used by Mihalcea and the second named author to prove an (equivariant) quantum Chevalley rule. In this paper we give a full description of the irreducible components of curve neighborhoods in terms of the Hecke product of (appropriate) Weyl group elements, $k$-strict partitions, and BC-partitions. The latter set of partitions respect the Bruhat order with inclusions. O…
We establish P=W and PI=WI conjectures for character varieties with structural group $\mathrm{GL}_n$ and $\mathrm{SL}_n$ which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and…
In this note we derive from deep results due to Clozel-Ullmo the density of Noether-Lefschetz loci inside the moduli space of marked (polarized) irreducible holomorphic symplectic (IHS) varieties. In particular we obtain the density of Hilbert schemes of points on projective $K3$ surfaces and of projective generalized Kummer varieties in their moduli spaces. We present applications to the existence of rational curves on projective deformations of such varieties, to the study of relevant cones of divisors, and a refinement of Hassett's result on cubic fourfolds whose Fano variety of lines is is…
Let $X={\bf P}^2\times{\bf P}^{n-1}$ embedded with $Ø(1,2)$. We prove that its $(n+1)$-secant variety $σ_{n+1}(X)$ is a hypersurface, while it is expected that it fills the ambient space. The equation of $σ_{n+1}(X)$ is the symmetric analog of the Strassen equation. When $n=4$ the determinantal map takes $σ_5(X)$ to the hypersurface of Lüroth quartics, which is the image of the Barth map studied by LePotier and Tikhomirov. This hint allows to obtain some results on the jumping lines and the Brill-Noether loci of symplectic bundles on ${\bf P}^2$ by using the higher secant varieties of $X$.
