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 $W=V^{\oplus n'} \oplus V*^{ \oplus n'}$, and $\mu: W \rightarrow Lie(G)*$ the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct canonical desingularizations of the irreducible components of the symplectic reduction $\mu^{-1}(0)//G$ for classes of examples where $G=GL(V)$, $O(V)$, or $Sp(V)$. Such components are isomorphic to closures of nilpotent orbits in simple Lie algebras and, for these classes of examples, we determine all the cases where the Hilbert-Chow morphism is a symplectic desingularization.

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

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

Ronan Terpereau. Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups. 2013. ⟨hal-00799369v2⟩