Pseudoconvex-Concave Duality and Regularization of Currents

Summary

We investigate some basic properties of Finsler metrics on holomorphic vector bundles, in the perspective of obtaining geometric versions of the Serre duality theorem. We establish a duality framework under which pseudoconvexity and pseudoconcavity properties get exchanged — up to some technical restrictions. These duality properties are shown to be related to several geometric problems, such as the conjecture of Hartshorne and Schneider, asserting that the complement of a q-codimensional algebraic subvariety with ample normal bundle is q-convex. In full generality, a functorial construction of Finsler metrics on symmetric powers of a Finslerian vector bundle is obtained. The construction preserves positivity of curvature, as expected from the fact that tensor products of ample vector bundles are ample. From this, a new shorter and more geometric proof of a basic regularization theorem for closed (1,1) currents is derived. The technique is based on the construction of a mollifier operator for plurisubharmonic functions, depending on the choice of a Finsler metric on the cotangent bundle and its symmetric powers.

Introduction

The goal of the present paper is to investigate some duality properties connecting pseudoconvexity and pseudoconcavity. Our ultimate perspective would be a geometric duality theory parallel to Serre duality, in the sense that Serre duality would be the underlying cohomological theory. Although similar ideas have already been used by several authors in various contexts — for example, for the study of direct images of sheaves [Ramis et al. 1971], or in connection with the study of Fantappie transforms and lineal convexity [Kiselman 1997], or in the study of Monge-Ampere equations [Lempert 1985]—we feel that the “convex-concave” duality theory still suffers from a severe lack of understanding.