Positivity of Schur forms for strongly decomposably positive vector bundles

TL;DR

This paper introduces two new types of strong decomposable positivity for vector bundles, establishes criteria for them, and proves positivity properties of their Schur forms, extending known results in complex geometry.

Contribution

It defines strongly decomposable positivity types, provides criteria, and proves positivity of Schur forms for these bundles, generalizing previous results.

Findings

Schur forms of type I are weakly positive

Schur forms of type II are positive

Answers Griffiths' question affirmatively for strongly decomposable bundles

Abstract

In this paper, we define two types of strongly decomposable positivity, which serve as generalizations of (dual) Nakano positivity and are stronger than the decomposable positivity introduced by S. Finski. We provide the criteria for strongly decomposable positivity of type I and type II and prove that the Schur forms of a strongly decomposable positive vector bundle of type I are weakly positive, while the Schur forms of a strongly decomposable positive vector bundle of type II are positive. These answer a question of Griffiths affirmatively for strongly decomposably positive vector bundles. Consequently, we present an algebraic proof of the positivity of Schur forms for (dual) Nakano positive vector bundles, which was initially proven by S. Finski.