We discuss a particular class of rational Gorenstein singularities, which we call symplectic. A normal variety V has symplectic singularities if its smooth part carries a closed symplectic 2-form whose pull-back in any resolution X --> V extends to a holomorphic 2-form on X . Our main result is the classification of isolated symplectic singularities with smooth projective tangent cone. They are in one-to-one correspondence with simple complex Lie algebras: to a Lie algebra g corresponds the singularity at 0 of the closure of the minimal (nonzero) nilpotent adjoint orbit in g .
We prove the uniqueness of crepant resolutions for some quotient singularities and for some nilpotent orbits. The finiteness of non-isomorphic symplectic resolutions for 4-dimenensional symplectic singularities is proved. We also give an example of symplectic singularity which admits two non-equivalent symplectic resolutions.
This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions, which could also serve as an introduction to this subject.
We give some necessary conditions for the existence of a symplectic resolution for quotient singularities. The McKay correspondence is also worked out for these resolutions.
We study symplectic resolutions of the Hilbert scheme of two points on a surface with one ADE-singularity. We also characterize such singularities by central fibers of their symplectic resolutions. As an application, we show that these singularities are isomorphic to the Slodowy slices which are transversal to the `sub-subregular' orbits in the nilpotent cones of ADE-types.
In this paper we shall prove that the singular locus of a symplectic singularity has no codimension 3 irreducible components. As a corollary, a symplectic singularity is terminal if and only if its singular locus has codimension $\geq 4$. It is hoped that a symplectic singularity has much stronger properties.
We consider symplectic singularities in the sense of A. Beauville as examples of Poisson schemes. Using Poisson methods, we prove that a symplectic singularity admits a finite stratification with smooth symplectic strata. We also prove that in the formal neighborhood of a closed point in some stratum, the singularity is a product of the stratum and a transversal slice. The transversal slice is also a symplectic singularity, and the product decomposition is compatible with natural Poisson structures. Moreover, we prove that the transversal slice admits a $C^*$-action dilating the symplectic for…
We develop a method of finding a Cox ring of a crepant resolution of a quotient singularity with a torus action and apply it to examples of symplectic quotient singularities in dimension 4. In addition we obtain a bound on the degrees of homogeneous generators of the Cox ring in a more general setup, based on the multigraded Castelnuovo-Mumford regularity and Kawamata-Viehweg vanishing.
For positive integers N and d, there are only finite number of conical symplectic varieties of dimension 2d with maximal weights N, up to isomorphism. The maximal weight of a conical symplectic variety X is, by definition, the maximal weight of the minimal homogeneous generators of the coordinate ring R of X.
We investigate Cox rings of symplectic resolutions of quotients of $\mathbb{C}^{2n}$ by finite symplectic group actions. We propose a finite generating set of the Cox ring of a symplectic resolution and prove that under a condition concerning monomial valuations it is sufficient. Also, three 4-dimensional examples are described in detail. Generators of the (expected) Cox rings of symplectic resolutions are computed and in one case a resolution is constructed as a GIT quotient of the spectrum of the Cox ring.
Let (X, ω) be an affine symplectic variety. Assume that X has a C^*-action with positive weights and ωis homogeneous with respect to the C^*-action. We prove that the algebraic fundamental group of the smooth locus X_{reg} is finite. This is a collorary to a more general theorem: If an affine variety X has a C^*action with positive weights and the log pair (X, 0) has klt singularities, then the algebraic fundamental group of X_{reg} is finite.
We consider the quotients $X = V/G$ of a symplectic complex vector space $V$ by a finite subgroup $G \subset Sp(V)$ which admit a smooth crepant resolution $Y \to X$. For such quotients, we prove the homological McKay correspondence conjectured by M. Reid. Namely, we construct a natural basis in the homology space $H_\cdot(Y,\Q)$ whose elements are numbered by the conjugacy classes in the finite group $G$.
We study the local symplectic algebra of the 1-dimensional isolated complete intersection singularity of type Sμ. We use the method of algebraic restrictions to classify symplectic Sμ singularities. We distinguish these symplectic singularities by discrete symplectic invariants. We also give the geometric description of them.
Let X be an affine normal variety with a C^*-action having only positive weights. Assume that X_{reg} has a symplectic 2-form w of weight l. We prove that, when l is not zero, the w is a unique symplectic 2-form of weight l up to C^*-equivariant automorphism When $l = 0$, we have a counter-example to this statement. In the latter half of the article, we associate to X a projective variety P(X) and prove that P(X) has a contact orbifold structure. Moreover, when X has canonical singularities, the contact orbifold structure is rigid under a small deformation. By using the contact structure on P(…
Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X \to V/G$, the bounded derived category $D^b(Coh(X))$ of coherent sheaves on $X$ is equivalent to the bounded derived category $D^b_G(Coh(V))$ of $G$-equivariant coherent sheaves on $V$.
We generalize a theorem of Delzant classifying compact connected symplectic manifolds with completely integrable torus actions to certain singular symplectic spaces. The assumption on singularities is that if they are not finite quotient then they are isolated.
We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t=0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.
We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $W_8$ and $W_9$ singularities. We use discrete symplectic invariants to distinguish symplectic singularities of the curves. We also give the geometric description of symplectic classes.
We describe a variety of symplectic surgeries (not a priori compatible with Kahler structures) which are obtained by combining local Kahler degenerations and resolutions of singularities. The effect of the surgeries is to replace configurations of Lagrangian spheres with symplectic submanifolds. We discuss several examples in detail, relating them to existence questions for symplectic manifolds with $c_1>0, c_1=0, c_1<0$ in four and six dimensions.
We study the local symplectic algebra of the 0-dimensional isolated complete intersection singularities. We use the method of algebraic restrictions to classify these symplectic singularities. We show that there are non-trivial symplectic invariants in this classification.
This short note is a supplement to the previous article with the same title. Here we treat a conical symplectic variety obtained as a finite covering of a (not necessarily normal) nilpotent orbit closure of a complex semisimple Lie algebra.
We prove that a conical symplectic variety with maximal weight 1 is isomorphic to one of the following: (i) an affine space with the standard symplectic form (ii) a nilpotent orbit closure of a complex semisimple Lie algebra with the Kirillov-Kostant form.
We prove that every minimal symplectic filling of the link of a quotient surface singularity can be obtained from its minimal resolution by applying a sequence of rational blow-downs and symplectic antiflips. We present an explicit algorithm inspired by the minimal model program for complex 3-dimensional algebraic varieties.
We investigate the Poisson geometry of the Marsden-Weinstein reductions of the moment map associated to the cotangent bundle of the space of representations of a quiver. We show that the stratification by representation type equals the stratification by symplectic leaves. The deformed symplectic quotient singularities - spectra of centres of symplectic reflection algebras associated to a wreath product - are shown to be isomorphic to reductions of a certain quiver. This establishes a method to calculate when these deformations are smooth. Furthermore, the isomorphism identifies symplectic leav…
Braverman, Finkelberg and Nakajima have recently given a mathematical construction of the Coulomb branches of a large class of $3d$ $\mathcal{N} =4$ gauge theories, as algebraic varieties with Poisson structure. They conjecture that these varieties have symplectic singularities. We confirm this conjecture for all quiver gauge theories without loops or multiple edges, which in particular implies that the corresponding Coulomb branches have finitely many symplectic leaves and rational Gorenstein singularities. We also give a criterion for proving that any particular Coulomb branch has symplectic…
We construct a new infinite family of 4-dimensional isolated symplectic singularities with trivial local fundamental group, answering a question of Beauville raised in 2000. Three constructions are presented for this family: (1) as singularities in blowups of the quotient of $\mathbb{C}^4$ by the dihedral group of order $2d$, (2) as singular points of Calogero-Moser spaces associated with dihedral groups of order $2d$ at equal parameters, (3) as singularities of a certain Slodowy slice in the $d$-fold cover of the nilpotent cone in ${\mathfrak{sl}}_d$.
We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.
Let G be a finite group acting on a symplectic complex vector space V. Assume that the quotient V/G has a holomorphic symplectic resolution. We prove that G is generated by "symplectic reflectionsd"', i.e. symplectomorphisms with fixed space of codimension 2 in V. Symplectic resolutions are always semismall. A crepant resolution of V/G is always symplectic. We give a symplectic version of Nakamura conjectures.
We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution has strong implications for the symplectic cohomology and conversely. We also use our computations to give a contact invariant of the link of the singularities and thereby distinguish many contact structures on connected sums of $S^2 \times S^3$.
Local symplectic contractions are resolutions of singularities which admit symplectic forms. Four dimensional symplectic contractions are (relative) Mori Dream Spaces. In particular, any two such resolutions of a given singularity are connected by a sequence of Mukai flops. We discuss the cone of movable divisors on such a resolution; its faces are determined by curves whose loci are divisors, we call them essential curves. The movable cone is divided into nef chambers which are related to different resolutions; this subdivision is determined by classes of 1-cycles. We also study schemes param…
In recent work, we introduced topological notions of simple normal crossings symplectic divisor and variety, showed that they are equivalent, in a suitable sense, to the corresponding geometric notions, and established a topological smoothability criterion for them. The present paper extends these notions to arbitrary normal crossings singularities, from both local and global perspectives, and shows that they are also equivalent to the corresponding geometric notions. In subsequent papers, we extend our smoothability criterion to arbitrary normal crossings symplectic varieties and construct a …
We study the hyperplane arrangements associated, via the minimal model programme, to symplectic quotient singularities. We show that this hyperplane arrangement equals the arrangement of CM-hyperplanes coming from the representation theory of restricted rational Cherednik algebras. We explain some of the interesting consequences of this identification for the representation theory of restricted rational Cherednik algebras. We also show that the Calogero-Moser space is smooth if and only if the Calogero-Moser families are trivial. We describe the arrangements of CM-hyperplanes associated to sev…
We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of adjoint orbits in a semisimple Lie algebra to the set of primitive ideals in the universal enveloping algebra. We show that the map is in…
Suppose that $C=(C_1,..., C_m)$ is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold $(X,ω)$ with connected, negative definite intersection graph $Γ_C$. We show that by replacing an appropriate neighborhood of $\cup C_i$ with a smoothing $W_S$ of a normal surface singularity $(S, 0)$ with resolution graph $Γ_C$, the resulting 4-manifold admits a symplectic structure. This operation generalizes the rational blow-down operation of Fintushel-Stern for other configurations, and therefore extends Symington's result about symplectic rational blow-downs.
Let $Γ$ be a finite subgroup of $\mathrm{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V / Γ$. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero-Moser space. In this way, we give a simple formula for the number of $\mathbb{Q}$-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik-Solomon algebra naturally associated to the Calogero-Moser deformation. This dimension is explici…
We introduce topological notions of normal crossings symplectic divisor and variety and establish that they are equivalent, in a suitable sense, to the desired geometric notions. Our proposed concept of equivalence of associated topological and geometric notions fits ideally with important constructions in symplectic topology. This partially answers Gromov's question on the feasibility of defining singular symplectic (sub)varieties and lays foundation for rich developments in the future. In subsequent papers, we establish a smoothability criterion for symplectic normal crossings varieties, in …
Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_0$ at level $0$ of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_0$. We show that if $(V, G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This in particular implies that the real symplectic quotient is graded Gorenstein. In the case that $K$ is a torus or $\operatorname{SU}_2$, we sho…
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.
Let Y be a normal projective variety and p a morphism from X to Y, which is a projective holomorphic symplectic resolution. Namikawa proved that the Kuranishi deformation spaces Def(X) and Def(Y) are both smooth, of the same dimension, and p induces a finite branched cover f from Def(X) to Def(Y). We prove that f is Galois. We proceed to calculate the Galois group G, when X is simply connected, and its holomorphic symplectic structure is unique, up to a scalar factor. The singularity of Y is generically of ADE-type, along every codimension 2 irreducible component B of the singular locus, by Na…
A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities $\tilde{E}_6,\tilde{E}_7$ and $\tilde{E}_8$. Our main application is to the classification of Po…
Recently, Herbig--Schwarz--Seaton have shown that $3$-large representations of a reductive group $G$ give rise to a large class of symplectic singularities via Hamiltonian reduction. We show that these singularities are always terminal. We show that they are $\mathbb{Q}$-factorial if and only if $G$ has finite abelianization. When $G$ is connected and semi-simple, we show they are actually locally factorial. As a consequence, the symplectic singularities do not admit symplectic resolutions when $G$ is semi-simple. We end with some open questions.
We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G\subset Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the rational cohomology H^*(X) of any smooth symplectic resolution X \to V/G (multiplicative McKay correspondence). We prove further that if G i…
We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity \C^n/G for a finite subgroup G in SL(n,\C) and any crepant resolution Y, we prove that the rank of positive symplectic cohomology SH_+(Y) is the number of conjugacy classes of G, and that twice the age grading on conjugacy classes is the \Z-grading on SH_+(Y) by the Conley-Zehnder index. The generalised McKay correspondence follows as SH_+(Y) is naturally isomorphic to ordinary cohomology H(Y), due to a vanishing result for full symplectic cohomology. In the Appendix we co…
Our previous papers introduce topological notions of normal crossings symplectic divisor and variety, show that they are equivalent, in a suitable sense, to the corresponding geometric notions, and establish a topological smoothability criterion for normal crossings symplectic varieties. The present paper constructs a blowup, a complex line bundle, and a logarithmic tangent bundle naturally associated with a normal crossings symplectic divisor and determines the Chern class of the last bundle. These structures hav applications in constructions and analysis of various moduli spaces. As a coroll…
Let X be a minimal surface of general type with positive geometric genus ($b_+ > 1$) and let $K^2$ be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length $\ell$ in a Q-Gorenstein degeneration, then $\ell \leq 4K^2 + 7$. This improves on the current best-known upper bound due to Lee ($\ell \leq 400(K^2)^4$). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball $B_{p,1}$…
