Bargmann–Fock extension from singular hypersurfaces

Abstract

Abstract We establish sufficient conditions for extension of weighted- L 2 ${L^{2}}$ holomorphic functions from a possibly singular hypersurface W to the ambient space ℂ n ${\mathbb{C}^{n}}$ . The L 2 ${L^{2}}$ -norms we use are the so-called generalized Bargmann–Fock norms, and thus there are restrictions on the singularities of W as well as the density of W. Our sufficient conditions are that W has density less than 1 and is uniformly flat in a sense that extends to singular varieties the notion of uniform flatness introduced in [5]. We present an example of Ohsawa showing that uniform flatness is not necessary for extension in the singular case, and find an example showing that, for rather different reasons, uniform flatness is also not necessary in the smooth case. The latter answers in the negative a question posed in [5].