Deformation theory of singular symplectic n-folds

By Y. Namikawa. 2000

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By a symplectic manifold (or a symplectic n-fold) we mean a compact Kaehler manifold of even dimension n with a non-degenerate holomorphic 2form ω, i.e. ω is a nowhere-vanishing n-form. This notion is generalized to a variety with singularities. We call X a projective symplectic variety if X is a normal projective variety with rational Gorenstein singularities and if the regular locus U of X admits a non-degenerate holomorphic 2-form ω. A symplectic variety will play an important role together with a singular Calabi-Yau variety in the generalized Bogomolov decomposition conjecture. Now that essentially a few examples of symplectic manifolds are discovered, it seems an important task to seek new symplectic manifolds by deforming symplectic varieties. In this paper we shall study a projective symplectic variety from a view point of deformation theory. If X has a resolution π : X → X such that (X, πω) is a symplectic manifold, we say that X has a symplectic resolution. Our first results are concerned with a birational contraction map of a symplectic manifold.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/79b2d4ccde1ab406a35e8b3804be5b8c269b4f9b

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Citations in the corpus

Symplectic involutions on deformations of K3[2] (2011, background): “Then isolated examples of such automorphisms were given by Namikawa [16], by Beauville [3] and later by Kawatani [11] and Amerik [1].”
Symplectic resolutions for nilpotent orbits (2002, background): “e → V such that codim(π−1(Sing(V))) ≥ 2. Proof. Let π : Ve → V be a symplecitc resolution. Suppose that the exceptional set E = π−1(Sing(V)) has a component of codimension 1, then by corollary 1.5 of [Nam] (where the proof works without the assumption “projective”) the image of this component by π i”
Remarks on degenerations of hyper-K\"ahler manifolds (2017, background): “The rigidity results of Namikawa [62, 63] then allow us to conclude that the degeneration can be modified to give a smooth family.”
Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections (2009, background): “is group Gis Z/2Z. Let Σ ⊂ Y be the singular locus. The dissident locus Σ0 ⊂ Σ is the locus along which the singularities of Y fail to be of ADEtype. Σ0 is a closed subvariety of Σ. Proposition 3.5. ([Na1], Propositions 1.6, [W]) Y has only canonical singularities. The dissident locus Σ0 has codimen”
Singular symplectic moduli spaces (2005, methodology): “A symplectic resolution of U is a resolution σ : U � → U of the singularities of U such that the restriction of the symplectic form on M s v to U reg extends to a symplectic form on U � . In such a case the morphism σ would have to be semismall according to a result of Namikawa, [ 27 ] Prop.”
The global moduli theory of symplectic varieties (2018, background): “Bogomolov{Fujiki form is the quadratic form q X := q X;˙. It is not hard now to deduce a local Torelli theorem for locally trivial deformations. Preliminary versions have been established by Namikawa [Nam01a], Kirschner [Kir15, Theorem 3.4.12], Matsushita [Mat15], and the authors [BL16]. Lemma 5.5 (”
On the monodromy of moduli spaces of sheaves on K3 surfaces II (2003, background): “ersal family and q: B→ B/ρ δ the quotient morphism. An aﬃrmative answer to Question 2.3 is known in the case of ρ δ, since the Hilbert-Chow morphism is a contraction of type A 1 as in Namikawa’s work [Nam]. The quotient B/ρ δ is the local deformation space of S(n), the generic ﬁber of the universal ”
A survey of Torelli and monodromy results for holomorphic-symplectic varieties (2011, unknown): “osition 6.2 relies heavily on Druel’s result stated above in Proposition 6.1. The fact that RE′ belongs to Mon2 Hdg(X ′) was proven earlier in ([Ma6], Theorem 1.4), using fundamental work of Namikawa [Nam1] (see [Nam3] for an alternative proof). The author does not know if the analogue of Propositio”
Poisson deformations of affine symplectic varieties (2006, background): “1.1) also works for the compact case. More exactly, let Y be a projective symplectic variety and let π : X →Y be a crepant (projective) resolution. We assume that h0(X,Ω2 X) = 1 and h1(X,O X) = 0. By [Na], the Kuranishi spaces Def(X) and Def(Y) are both non-singular, and the induced map f¯: Def(X) →”
Intersection Numbers of Extremal Rays on Holomorphic Symplectic Varieties (2009, background): “Namikawa [26]) that W◦ is holomorphic symplectic and W \ W◦ has even codimension in Y .”