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Yoshinori Namikawa

Publications and source records attributed to Yoshinori Namikawa.

32 records · Page 2Linked to original sources
arXiv2008

Induced nilpotent orbits and birational geometry

In general, a nilpotent orbit closure in a complex simple Lie algebra \g, does not have a crepant resolution. But, it always has a Q-factorial terminalization by the minimal model program. According to B. Fu, a nilpotent orbit closure has a crepant resolution only when it is a Richardson orbit, and the resolution is obtained as a Springer map for it. In this paper, we shall generalize this result to Q-factorial terminalizations when \g$ is classical. Here, the induced orbits play an important role instead of Richardson orbits.

math.AG↗
arXiv2007

Birational geometry and deformations of nilpotent orbits

This is a continuation of math.AG/0408274, where we have described the relative movable cone for a Springer resolution of the closure of a nilpotent orbit in a complex simple Lie algebra. But, in general, the movable cone does not coincide with the whole space of numerical classes of divisors on the Springer resolution. The purpose of this paper is, to describe the remainder. We shall first construct a deformation of the nilpotent orbit closure in a canonical manner according to Brieskorn and Slodowy, and next describe all its crepant simultaneous resolutions. This construction enables us to divide the whole space into a finite number of chambers. Moreover, by using this construction, one can generalize the main result of math.AG/0408274 to arbitrary Richardson orbits whose Springer maps have degree > 1. New Mukai flops, different from those of type A,D,E_6, will appear in the birational geometry for such orbits.

math.AG↗
arXiv2006

Flops and Poisson deformations of symplectic varieties

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.

math.AG↗
arXiv2005

On deformations of Q-factorial symplectic varieties

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 minimal model conjecture were true.

math.AG↗
arXiv2005

Birational geometry of symplectic resolutions of nilpotent orbits II

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 E_6) in connecting symplectic resolutions. In particular, Mukai flops of type E_6 are new. All arguments of Part I : math.AG/0404072 which use flags, are replaced by those which use only Dynkin diagrams.

math.AG↗
arXiv2004

Birational Geometry of symplectic resolutions of nilpotent orbits

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 Springer resolutions via the cotanegent bundles T^*G^+ and T^*G^-, which is the Mukai flop of type D. Our result would clarify the geometric meaning of the results of Spaltenstein and Hesselink. To illustrate this, three examples will be given. Another purpose of this paper is to give an affimative answer to a conjecture in the paper "Uniqueness of crepant resolutions and symplectic singularities" math.AG/0306091, for nilpotent orbit cases. A key idea is the notion of a "Dixmier sheet" studied by Borho and Kraft.

math.AG↗
arXiv2003

Mukai flops and derived categories II

This paper is a sequel to math.AG/0203287. A generalization of the Mukai flop has been studied by E. Markman. Here we call it a stratified Mukai flop. In this paper, we observe that, for a stratified Mukai flop: $X \to \bar{X} \leftarrow X^+$, the fiber product $X \times_{\bar{X}}X^+$ defines an isomorphism of Grothendieck groups $K(X) \to K(X^+)$, but, at the level of derived categories, the fiber product does not generally give an equivalence.

math.AG↗
arXiv2003

Uniqueness of crepant resolutions and symplectic singularities

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.

math.AG↗
arXiv2002

Counter-example to global Torelli problem for irreducible symplectic manifolds

We shall give an example of irreducible symplectic manifolds X and Y which are not bimeromorphic, but have the same periods in the second cohomologies. More explicitly, X and Y are generalized Kummer varieties of dim 4 for a general complex torus T of dim 2 and its dual torus T^*. A key result is an observation of Shioda for the periods of T and T^*. As a concluding remark, we shall discuss the difference between our case and Kummer surfaces.

math.AG↗
arXiv2002

Mukai flops and derived categories

In this note, we shall prove that two smooth projective varieties of dim 2n connected by a Mukai flop have equivalent bounded derived categories. More precisely, let $ϕ: X - - \to X^+$ be a Mukai flop with centers $Y \subset X$ and $Y^+ \subset X^+$. In our case, the natural fuctor $Φ: D(X) \to D(X^+)$ defined by the graph of $ϕ$ is not fully faithful. Instead, let $X \to {\bar X}$ and $X^+ \to {\bar X}$ be the birational contraction maps of the centers, and put ${\hat X} := X \times_{{\bar X} X^+$. Then ${\hat X}$ is a normal crossing variety with two irreducible components. This ${\hat X}$ defines a functor $Ψ: D(X) \to D(X^+)$. We shall prove that this $Ψ$ is an equivalence. Recently, Wierzba and Wisniewski have announced that two birationally equivalent, complex projective symplectic 4-folds are connected by a finite sequence of Mukai flops. Our result with this shows that $D(X)$ is a birational invariant for complex projective symplectic 4-folds.

math.AG↗
arXiv2001

Deformation theory of singular symplectic n-folds

The notion of a holomorphically symplectic manifold can be generalized to the singular one. This paper studies the birational contraction maps between symplectic varieties, and then describes the deformation of a symplectic variety which has a symplectic resolution. Some examples are also included.

math.AG↗
arXiv2001

Extension of 2-forms and symplectic varieties

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.

math.AG↗
arXiv2001

A note on symplectic singularities

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.

math.AG↗