Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

Ronan Terpereau (1)

1 IF - Institut Fourier

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$, and let $\mu:\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$, $O(V)$, or $Sp(V)$. For these classes of examples, $\mu^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert-Chow morphism with the (well-known) symplectic desingularizations of $\mu^{-1}(0)/\!/G$.

Mots clés: symplectic singularity; Invariant Hilbert scheme; nilpotent orbit; homogeneous bundle; desingularization

Domaines: Géométrie algébrique [math.AG]; Géométrie symplectique [math.SG]

Ronan Terpereau. Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups. Mathematische Zeitschrift, 2014, 277 (1-2), pp.339-359. ⟨hal-00799369v3⟩