1eiel dz i S dz li . The Z has singularities, and Z has no symplectic resolution when l b 1.(iii) These are symplectic varieties studied by O'Grady [O]. Let S be a polarized K3 surface. Let c be an even number with c f 4. Denote by M 0Y c the moduli space of rank 2 Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/1/15 3:13 AM semi-stable torsion free sheaves with c 1 0 and c 2 c. M 0Y c becomes a projective symplectic variety of dim 4c À 6. The singular locus has dimension 2c. Moreover, O'Grady showed that M 0Y 4 has a symplectic resolution, however M 0Y c has Q-factorial terminal singularities when c f 6 (cf. section 3 of [O]). Therefore M 0Y c have no symplectic resolution when c f 6.A symplectic singularity/variety will play an important role in the generalized Bogomolov decomposition conjecture (cf.[Kata], [Mo]):where Y 1 is an Abelian variety, Y 2 is a symplectic variety, and Y 3 is a Calabi-Yau variety.In this conjecture we hope that it is possible to replace Y 2 and Y 3 by their birational models with only Q-factorial terminal singularities respectively. Main results are these.Theorem 7 (Stability Theorem). Let ZY o be a projective symplectic variety. Let g: Z 3 h be a projective¯at morphism from Z to a 1-dimensional unit disc h with g À1 0 Z. Then o extends sideways in the¯at family so that it gives a symplectic 2-form o t on each ®ber Z t for t e h e with a su½ciently small e.In the above, the result should also hold for a (non-projective) symplectic variety ZY o and for a proper¯at morphism g. But two ingredients remained unproved in the general case (cf. Remark below Theorem 7).Let Z be a symplectic variety. Put X SingZ and U X Zn. Let p: Z 3 S be the Kuranishi family of Z, which is, by de®nition, a semi-universal¯at deformation of Z with p À1 0 Z for the reference point 0 e S. When codim r Z f 4, S is smooth by [Na 1], Theorem 2.5. Z is not projective over S. But we can show that every member of the Kuranishi family is a symplectic variety (cf. Theorem 7 H ). De®ne U to be the locus in Z where p is a smooth map and let p: U 3 S be the restriction of p to U. Then we haveCodim r Z f 4X Then the following hold.(1) R 2 p à p À1 O S is a free O S module of ®nite rank. Let H be the image of the composite R 2 p à C 3 R 2 p à C 3 R 2 p à p À1 O S . Then H is a local system on S with H s H 2 U s Y C for s e S.(2) The restriction map H 2 ZY C 3 H 2 UY C is an isomorphism. Take a resolution n:Z 3 Z in such a way that n À1 U q U. For e H 2 UY C we take a lift e H 2 ZY C by the composite H 2 UY C q H 2 ZY C 3 H 2 ZY C. Choose o e H 0 UY 2 U C. This o Namikawa, 2-forms and symplectic varieties 124 Brought to you by |
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.Comment: Version 7(36 pages
Proposition (1.4).Let π :X → X be a birational projective morphism from a projective symplectic n-foldX to a normal n-fold X. Let S i be the set of points p ∈ X such that dim π −1 (p) = i. Then dim S i ≤ n − 2i. In particular, dim π −1 (p) ≤ n/2. Proposition (1.6). Let π :X → X be a birational projective morphism from a projective symplectic n-foldX to a normal n-fold X. Then X has only canonical singularities and its dissident locus Σ 0 has codimension at least 4 in X. Moreover, if Σ \ Σ 0 is non-empty, then Σ \ Σ 0 is a disjoint union of smooth varieties of dim n − 2 with everywhere non-degenerate 2-forms.When X has only an isolated singularity p ∈ X, every irreducible component of π −1 (p) is Lagrangian (Proposition (1.11)). In this situation it is conjectured Birational contraction maps of symplectic n-foldsA symplectic n-fold means a symplectic manifold of dimension n. We shall state three lemmas which will be used later. The first lemma is essentially a linear algebra. (1.1). Let V be a complex manifold with dim V = 2r and let ω be an everywhere non-degenerate holomorphic 2-form on V (i.e. ∧ r ω is nowherevanishing.) Let E be a subvariety of V with dim E > r. Then ω| E is a non-zero 2-form on E. LemmaLemma (1.2). Let f : V → W be a birational projective morphism from a complex manifold V to a normal variety W . Let p ∈ W and assume that the germ (W, p) of W at p has rational singularities. Assume that E := f −1 (p) is a simple normal crossing divisor of V . Then H 0 (E,Ω i E ) = 0 for i > 0, wherê Ω i E := Ω i E /(torsion).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.