$L^2$-minimal extensions over Hermitian symmetric domains

Abstract

In this paper, we study the $L^2$-minimal extension problem for polarized variations of Hodge structures over Hermitian symmetric domains. We are able to explicitly find the $L^2$-minimal extensions using a group-theoretic construction. In particular, this gives a construction without using $L^2$-estimates as in the Ohsawa-Takegoshi type extension theorems. The key ingredient is the Harish-Chandra embedding of Hermitian symmetric domains. The construction of holomorphic sections might be of independent interest since it gives a concrete description in the setting of Hermitian VHS.