On Q-factorial terminalizations of nilpotent orbits

Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras

Let X be a Gorenstein normal 3-fold satisfying (ELF) with local rings which are at worst isolated hypersurface (e.g. terminal) singularities. By using the singular derived category D_{sg}(X) and its idempotent completion, we give necessary and sufficient categorical conditions for X to be Q-factorial and complete locally Q-factorial respectively. We then relate this information to maximal modification algebras(=MMAs), introduced in [IW10], by showing that if an algebra A is derived equivalent to X as above, then X is Q-factorial if and only if A is an MMA. Thus all rings derived equivalent to …

环面超Kähler簇与Q-因子化终端化

Toric hyperkähler varieties and Q-factorial terminalizations

本文研究了环面超Kähler簇的Q-因子化终端化问题，通过将$Y(A,0)$实现为另一环面超Kähler簇以实现Q-因子化，并给出了其具有crepant解消的充要条件及显式构造了通用Poisson形变。

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}…

Derived equivalences for Symplectic reflection algebras

In this paper we study derived equivalences for Symplectic reflection algebras. We establish a version of the derived localization theorem between categories of modules over Symplectic reflection algebras and categories of coherent sheaves over quantizations of Q-factorial terminalizations of the symplectic quotient singularities. To do this we construct a Procesi sheaf on the terminalization and show that the quantizations of the terminalization are simple sheaves of algebras. We will also sketch some applications: to the generalized Bernstein inequality and to perversity of wall crossing fun…

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.

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.