Note on the semi-continuity of the algebraic dimension

TL;DR

This paper proves that the direct image sheaf associated with an analytic family of compact complex manifolds is locally free, leading to an improved semi-continuity result for the algebraic dimension in such families.

Contribution

It establishes the local freeness of the direct image sheaf R^1 π_* (O_X) for analytic families, enhancing semi-continuity results for algebraic dimension.

Findings

The sheaf R^1 π_* (O_X) is locally free over the base space.

Improved semi-continuity theorem for algebraic dimension in analytic families.

Application to complex geometry and deformation theory.

Abstract

In this short Note we show that the direct image sheaf R 1 $\pi$ * (O X) associated to an analytic family of compact complex manifolds $\pi$ : X $\rightarrow$ S parametrized by a reduced complex space S is a locally free (coherent) sheaf of O S --modules. This result allows to improve a semi-continuity type result for the algebraic dimension of compact complex manifolds in an analytic family given in [B.15]. AMS Classification 2010. 32G05-32A20-32J10.