Desingularization of Hyperkähler Varieties I Desingularization of Singular Hyperkähler Varieties I

• In Section 4, we define locally homogeneous singularities. A space with locally homogeneous singularities (SLHS) is an analytic space X such that for all x ∈ X , the x-completion of a local ring OxX is isomorphic to an x-completion of associated graded ring (OxX)gr. We show that a complex variety is SLHS if and only if the underlying real analytic variety is SLHS. This allows us to define invariantly the notion of a hyperkähler SLHS. The natural examples of hyperkähler SLHS include the moduli spaces of stable holomorphic bundles, considered in [V-bun]. 2 We conjecture that every hyperkähler variety is a space with locally homogeneous singularities.