[Paper Review] Birational geometry for the covering of a nilpotent orbit closure

Yoshinori Namikawa|arXiv (Cornell University)|Jul 17, 2019

Advanced Algebra and Geometry20 references4 citations

TL;DR

This paper constructs an explicit Q-factorial terminalization for the universal cover of a nilpotent orbit closure in classical Lie algebras by using generalized Springer maps and parabolic induction. It provides a systematic method to build such resolutions via Levi subalgebras and finite covers, generalizing earlier results for the normalization of orbit closures and yielding explicit geometric realizations of crepant resolutions when nonsingular.

ABSTRACT

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.

Motivation & Objective

• To construct an explicit Q-factorial terminalization for the universal cover $X$ of a nilpotent orbit $O$ in a classical complex simple Lie algebra $\mathfrak{g}$.
• To extend known results on terminalizations of orbit closure normalizations to the case of nontrivial finite covers arising from the universal covering of $O$.
• To provide a group-theoretic, geometric construction of such terminalizations using parabolic subgroups and induced nilpotent orbits.
• To establish conditions under which the resulting resolution is nonsingular, thus yielding a crepant projective resolution.
• To generalize the framework of [Lo] and [Ma] to explicitly realize the finite cover $\pi: X \to \bar{O}$ via a generalized Springer map construction.

Proposed method

• Use parabolic induction: for a parabolic subgroup $Q \subset G$ with Levi decomposition $Q = U \cdot L$, and a nilpotent orbit $O' \subset \mathfrak{l}$, induce $O = \mathrm{Ind}^\mathfrak{g}_\mathfrak{l}(O')$.
• Construct the generalized Springer map $\mu: G \times^Q (\mathfrak{n} + \bar{O}') \to \bar{O}$, which is generically finite and $G$-equivariant.
• Lift the $Q$-action on $\mathfrak{n} + \bar{O}'$ to a $Q$-action on $\mathfrak{n} + X'$, where $X'$ is a finite cover of $\bar{O}'$ associated to an étale cover of $O'$.
• Form the space $Y = G \times^Q (\mathfrak{n} + X')$, and define $Z$ as the Stein factorization of $\mu \circ \pi'$, so that $Z \to \bar{O}$ is isomorphic to the original cover $\pi: X \to \bar{O}$.
• Show that the map $Y \to X$ is a Q-factorial terminalization by verifying $K_Y = \pi^* K_X$ and that $Y$ has Q-factorial terminal singularities.
• Use double induction (types I and II) on partitions to reduce the problem to known cases, particularly for $\mathfrak{so}(m)$, and construct explicit covers via kernel subgroups like $H = \ker[\sum: (\mathbb{Z}/2\mathbb{Z})^{\oplus 3} \to \mathbb{Z}/2\mathbb{Z}]$.

Experimental results

Research questions

• RQ1 Can a Q-factorial terminalization be explicitly constructed for the universal cover $X$ of a nilpotent orbit $O$ in a classical Lie algebra $\mathfrak{g}$?
• RQ2 Under what conditions does the generalized Springer map construction yield a terminalization of $X$?
• RQ3 How can the $\mathbf{C}^*$-action on $X$ be lifted to ensure weight 2 for the symplectic form $\omega$?
• RQ4 When is the constructed terminalization nonsingular, thus giving a crepant projective resolution?
• RQ5 What is the precise relationship between the degree of the cover $\pi: X \to \bar{O}$ and the degrees of the maps in the generalized Springer construction?

Key findings

• A Q-factorial terminalization of $X$, the universal cover of a nilpotent orbit closure $\bar{O}$, is explicitly constructed using parabolic induction and generalized Springer maps.
• The construction works for all classical Lie algebras and applies to all nilpotent orbits, including those with nontrivial fundamental group.
• The terminalization $Y \to X$ is given by $Y = G \times^Q (\mathfrak{n} + X')$, where $X'$ is a finite cover of a Levi orbit closure with $Q$-action lifting to $\mathfrak{n} + X'$.
• In the case where $Y$ is nonsingular, the map $Y \to X$ is a crepant projective resolution, generalizing earlier results for the normalization $\tilde{O}$.
• For $\mathfrak{so}(m)$, the method uses double inductions of types I and II to reduce to base cases, and the degree of the cover $\pi$ matches the product of degrees from intermediate maps.
• Explicit examples are provided: for $\bar{O}_{[15,8^2,3]} \subset \mathfrak{so}(34)$, the terminalization is $\mathrm{Spin}(34) \times^Q (\mathfrak{n} + X_{[2]} \times X_{[2^3]} \times X_{[7,4^2,3]}/H)$ with $H = \ker[\sum: (\mathbb{Z}/2\mathbb{Z})^3 \to \mathbb{Z}/2\mathbb{Z}]$.