Bemerkungen zur Deformationstheorie nichtrationaler Singularitäten

Abstract

Let$\tilde \pi :\tilde Z$ be a 1-convex holomorphic mapping between complex spaces$\tilde Z$ resp.S, and let$\tilde \pi = \pi \circ \sigma$ be the blowingdown factorization of$\tilde \pi$ over S. We prove in part 1 of the present note: The fiber π⁻¹(s₀) over a point s₀∈S is the Remmert quotient of$\tilde \pi ^{ - 1}$ if and only if every holomorphic function on$\tilde \pi ^{ - 1}$ (defined in a neighborhood of the exceptional subvariety of that fiber) can be extended holomorphically to$\tilde Z$. This is true, for instance, in the case:$\tilde \pi ^{ - 1}$ flat, S reduced at s₀ and dim$H^1 (\tilde \pi ^{ - 1} (s),\mathcal{O}(\tilde \pi ^{ - 1} (s))$, =const for all s∈S. In part 2, we use this result to obtain the following: For any Riemann surface R with genus g⩾2 there exists a 2-dimensional normal complex analytic singularity X such that the minimal resolution$\tilde X$ of X contains R as exceptional subvariety, and$\tilde X$ has a deformation over the unit disc S={|s|<1} which can not be blown down to a deformation of X.