Yoshinori Namikawa

Deformation theory of singular symplectic n-folds

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.

Global smoothing of Calabi-Yau threefolds.

Yoshinori Namikawa I,*, J.H.M. Steenbrink 2 I Department of Mathematics, Sophia University, Kioi-Cho, Tokyo 102, Japan 2 Mathematical Institute, University of Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands Oblatum 10-X-1994 & 13-VI-1995 Introduction Friedman [Fr] has studied the relationship between local and global deforma- tions of a threefold Z with isolated hypersurface singularities which admits small resolutions. One of his main results is as follows. Let Z be a Moishezon threefold with only ordinary double points { Pl ..... p. }. Assume that the canon- ical line bundle

ON DEFORMATIONS OF CALABI-YAU 3-FOLDS WITH TERMINAL SINGULARITIES

BY a Q-Calabi-Yau 3-fold X we mean a projective 3-fold X with only terminal singularities such that mK, + 0 for a positive integer m. When K, - 0, X is called a Calabi-Yau 3-fold. Every smooth projective 3-fold Z with K(Z) = 0 is birational to a Q-Calabi-Yau 3-fold (cf. [17], [7]). In this paper we first study the Kuranishi space of a Q-Calabi-Yau 3-fold and next study the behaviour of ample cones of Calabi-Yau 3-folds with terminal singularities under deformation. Our first result is the following.

On deformations of ℚ-factorial symplectic varieties

Abstract Our purpose is to give a positive answer to the following problem posed in [Namikawa, Y., Extension of 2-forms and symplectic varieties, J. reine angew. Math. 539 (2001), 123– 147.]: Problem. Let Z be a ℚ-factorial projective symplectic variety with terminal singularities. Assume that Z is smoothable by a suitable flat deformation. Is Z then non-singular from the first? We actually prove more: Main Theorem. Let Z be a ℚ-factorial projective symplectic variety with terminal singularities. Then any flat deformation of Z is locally trivial; in other words, it preserves all singularities on Z.

PROJECTIVITY CRITERION OF MOISHEZON SPACES AND DENSITY OF PROJECTIVE SYMPLECTIC VARIETIES

A Moishezon manifold is a projective manifold if and only if it is a Kahler manifold [13]. However, a singular Moishezon space is not generally projective even if it is a Kahler space [14]. Vuono [19] has given a projectivity criterion for Moishezon spaces with isolated singularities. In this paper we shall prove that a Moishezon space with 1-rational singularities is projective when it is a Kahler space (Theorem 1.6). We shall use Theorem 1.6 to show the density of projective symplectic varieties in the Kuranishi family of a (singular) symplectic variety (Theorem 2.4), which is a generalization of the result by Fujiki [4, Theorem 4.8] to the singular case. In the Appendix we give a supplement and a correction to the previous paper [15] where singular symplectic varieties are dealt with.

Poisson deformations of affine symplectic varieties, II

Let Y be an affine symplectic variety of dimension 2n, and let π : X → Y be a crepant resolution. By the definition, there is a symplectic 2-form σ̄ on the smooth part Yreg ∼= π (Yreg), and it extends to a 2-form σ on X . Since π is crepant, σ is a symplectic 2-form on X . The symplectic structures on X and Y define Poisson structures on them in a natural manner. One can define a Poisson deformation of X (resp. Y ) (cf. [Na 1]). A Poisson deformation of X is equivalent to a symplectic deformation, namely a deformation of the pair (X, σ). Let PDY : (Art)C → (Set)

Stratified local moduli of Calabi–Yau threefolds

Abstract We define a natural stratification on the Kuranishi space of Calabi–Yau threefold with terminal singularities. By using this stratification, we give a sufficient and necessary condition for a singular Calabi–Yau threefold to be smoothed by a flat deformation.

A characterization of nilpotent orbit closures among symplectic singularities

We prove that a conical symplectic variety with maximal weight 1 is isomorphic to (i) an affine space with the standard symplectic form, or (ii) a normal nilpotent orbit closure of a complex semisimple Lie algebra.

SLODOWY SLICES AND UNIVERSAL POISSON DEFORMATIONS

We classify the nilpotent orbits in a simple Lie algebra for which the restriction of the adjoint quotient map to a Slodowy slice is the universal Poisson deformation of its central fibre. This generalises work of Brieskorn and Slodowy on subregular orbits. In particular, we find in this way new singular symplectic hypersurfaces of dimension 4 and 6.

Poisson deformations of affine symplectic varieties

This is a continuation of math.AG/0609741. Let Y be an affine symplectic variety with a C^*-action with positive weights, and let \pi: X -> Y be its crepant resolution. Then \pi induces a natural map PDef(X) -> PDef(Y) of Kuranishi spaces for the Poisson deformations of X and Y. In the Part I, we proved that PDef(X) and PDef(Y) are both non-singular, and this map is a finite surjective map. In this paper (Part II), we prove that it is a Galois covering. Markman already obtained a similar result in the compact case, which was a motivation of this paper. As an application, we shall construct explicitly the universal Poisson deformation of the normalization \tilde{O} of a nilpotent orbit closure \bar{O} in a complex simple Lie algebra when \tilde{O} has a crepant resolution.