Curvature and relative volume forms

TL;DR

The work develops a metric framework for studying curvature of higher direct images in semistable families, linking Berndtsson-type positivity to the Kodaira–Spencer class and to Massey products. It introduces a curvature interpretation of liftability for Massey products via a quotient bundle $\mathcal{Q}$ and derives criteria that separate isotrivial from strongly non-isotrivial fibrations, including a concrete surface case with a positive index bound. The paper further establishes conditions under which fibrations become trivial on a Zariski open set, leveraging unitary-flat and ample components from Fujita-type decompositions. Overall, it connects analytic curvature techniques with algebraic geometry invariants (Kodaira–Spencer, Fujita decomposition, Massey products) to obtain new isotriviality and triviality criteria for families of varieties.

Abstract

Using metric techniques introduced by Berndtsson, we show a result on constancy of families dominated by a constant variety and, on the opposite side, a result on the strong non isotriviality of certain families of surfaces with positive index. We also give metric interpretations of liftability of relative volume forms and of strong non isotriviality in terms of the complex conjugate of a suitable representative of the Kodaira-Spencer class.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5