Extension of Holomorphic Functions Defined on Non Reduced Analytic Subvarieties

The goal of the talk will be to discuss L" extension properties of holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results are derived from L" approximation techniques, and they hold under (probably) optimal curvature conditions.