Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

Ronan Terpereau (1)

1 IF - Institut Fourier

Let G be a classical subgroup of GL(V) acting on the symplectic vector space W=n'V \oplus n'V*, 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). It is known that such components are isomorphic to closures of nilpotent orbits in simple Lie algebras, and 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-00799369v1⟩