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Yoshinori Namikawa

Publications and source records attributed to Yoshinori Namikawa.

At least 19 recordsLinked to original sources
arXiv2026

Towards a characterization of toric hyperkähler varieties among symplectic singularities II

This is a continuation of arXiv: 2408.03012. We answer affirmatively Question 5.10 posed in the previous article. More precisely, let $(X, ω)$ be a conical symplectic variety of dimension $2n$ with $wt(ω) = 2$, which has a projective symplectic resolution. Assume that $X$ admits an effective Hamiltonian action of an $n$-dimensional algebraic torus $T^n$, compatible with the conical $\mathbf{C}^*$-action. Then we prove that there is a $T^n$-equivariant algebraic isomorphism $(X, ω) \cong (Y(A,0), ω_{Y(A,0)})$ for a toric hyperkahler variety $Y(A, 0)$ with $A$ unimodular.

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arXiv2026

Canonical torus action on symplectic singularities

We show that any symplectic singularity lying on a smoothable projective symplectic variety locally admits a good action of $(\mathbb{C}^*)^r$, which is canonical. Under mild assumptions, we actually prove such singularity germ is the cone vertex over a contact orbifold with weak Kähler-Einstein metric, forcing $r=1$. In particular, it admits a (canonical) good $\mathbb{C}^*$-action, which also extends to (canonical) actions of $\mathbb{H}^*\supset SU(2)$. These settle Kaledin's conjecture conditionally but in a substantially stronger form by establishing the canonicity, the extensibility of the action, for instance. Our key idea is to use the Donaldson-Sun theory on local Kähler metrics in complex differential geometry to connect with the theory of Poisson deformations of symplectic varieties. For general symplectic singularities, we prove the same assertions, assuming that the Donaldson-Sun theory extends to such singularities along with suitable singular (hyper)Kähler metrics. Conversely, our results can also be used to study the local behavior of such metrics around the germ.

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arXiv2025

A remark on isolated symplectic singularities with trivial local fundamental group

Recently, Bellamy et al. constructed an infinite series of 4-dimensional isolated symplectic sngularities with trivial local fundamental group, inspired by a question of Beauville. In this short note, we introduce an easy construction of isolated symplectic singularities (of any dimension) with trivial local fundamental group. We will use a toric hyperkaehler construction.

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arXiv2025

Towards a characterization of toric hyperkähler varieties among symplectic singularities

Let $(X, ω)$ be a conical symplectic variety of dimension $2n$ which has a projective symplectic resolution. Assume that $X$ admits an effective Hamiltonian action of an $n$-dimensional algebraic torus $T^n$, compatible with the conical $\mathbf{C}^*$-action. A typical example of $X$ is a toric hyperkahler variety $Y(A,0)$. In this article, we prove that this property characterizes $Y(A,0)$ with $A$ unimodular. More precisely, if $(X, ω)$ is such a conical symplectic variety, then there is a $T^n$-equivariant (complex analytic) isomorphism $φ: (X, ω) \to (Y(A,0), ω_{Y(A,0)})$ under which both moment maps are identified. Moreover $φ$ sends the center $0_X$ of $X$ to the center $0_{Y(A,0)}$ of $Y(A,0)$.

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arXiv2023

Toric hyperkähler varieties and Q-factorial terminalizations

A toric hyperkähler variety is determined by combinatorial data A and $α$. Here A is an integer valued matrix and $α$ is a character of an algebraic torus $T^d$. $Y(A, α)$ is a crepant partial resolution of an affine toric hyperkähler variety $Y(A,0)$. However, $Y(A, α)$ is not generally a Q-factorial terminalization of $Y(A,0)$ even if $α$ is generic. In this article, we realize $Y(A,0)$ as another toric hyperkähler variety $Y(A^{\sharp}, 0)$ so that $Y(A^{\sharp}, α^{\sharp})$ is a Q-factorialization of $Y(A^{\sharp}, 0)$ for a generic $α^{\sharp}$. As an application, we give a necessary and sufficient condition for $Y(A,0)$ to have a crepant resolution. Moreover, we construct explicitly the universal Poisson deformation of $Y(A,0)$ in terms of $A^{\sharp}$.

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arXiv2022

Birational geometry for the covering of a nilpotent orbit closure

A nilpotent orbit $O$ of a complex semisimple Lie algebra $\mathfrak{g}$ has finite fundamental group. Associated with an etale cover of $O$, we have a finite cover of the closure $\bar{O}$ of $O$. In this article we consider the finite cover X associated with the universal cover of a nilpotent orbit $O$ of a classical simple Lie algebra $\mathfrak{g}$. We construct explicitly a Q-factorial terminalization of X in a group theoretic way.

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arXiv2022

Birational geometry for the covering of a nilpotent orbit closure II

Let $O$ be a nilpotent orbit of a complex semisimple Lie algebra $\mathfrak{g}$ and let $π: X \to \bar{O}$ be the finite covering associated with the universal covering of $O$. In the previous article we have explicitly constructed a $\mathbf{Q}$-factorial terminalization $\tilde{X}$ of $X$ when $\mathfrak{g}$ is classical. In the present article, we count how many different $\mathbf{Q}$-factorial terminalizations $X$ has. We construct the universal Poisson deformation of $\tilde{X}$ over $H^2(\tilde{X}, \mathbf{C})$ and look at the action of the Weyl group $W(X)$ on $H^2(\tilde{X}, \mathbf{C})$. The main result is an explicit geometric description of $W(X)$.

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arXiv2015

A finiteness theorem on symplectic singularities

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.

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arXiv2014

Poisson deformations and birational geometry

Let π: Y -> X be a crepant projective resolution of an affine symplectic variety X with a good C^*-action. We interpret the second cohomology H^2(Y, C) in two ways. First, H^2(Y, C) is the Picard group of Y tensorised with C. By the ample cones of different crepant resolutions of X, there is a natural chamber structure in H^2(Y, C). The second interpretation of H^2(Y, C) is the base space of the universal Poisson deformation $\mathcal Y$ of Y. Let D \subset H^2(Y, C) be the locus where the corresponding Poisson varieties are not affine. Then D is the union of finite number of hyperplanes, which gives a chamber structure in H^2(Y, C). These two chamber structures coincide.

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arXiv2013

Fundamental groups of symplectic singularities

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.

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arXiv2013

Equivalence of symplectic singularities

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(X), we discuss the equivalence problem for (X, w) up to contant. In most examples the symplectic structures turn out to be unique up to constant with very few exceptions.

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arXiv2010

Birational geometry for nilpotent orbits

This is a survey article prepared for the submission to "Handbook of moduli". The following topics are discussed: (i) Basic facts and examples of resolutions for nilpotent orbit (ii) Q-factorial terminalizations of nilpotent orbit closures and related birational geometry (iii) Poisson deformations of nilpotent orbit closures.

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arXiv2010

Poisson deformations of affine symplectic varieties

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.

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arXiv2010

Poisson deformations of affine symplectic varieties II

This is a continuation of math.AG/0609741. Let Y be an affine symplectic variety with a C^*-action with positive weights, and let π: X -> Y be its crepant resolution. Then πinduces 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.

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