On characteristic forms of positive vector bundles, mixed discriminants and pushforward identities

TL;DR

This paper proves positivity results for Schur polynomials in Chern forms of certain positive vector bundles, introduces new pushforward identities for characteristic forms, and explores limitations in characterizing positivity via Schur polynomials.

Contribution

It establishes differential-geometric positivity results for Schur polynomials, introduces refined pushforward identities, and analyzes the limitations of using Schur polynomials to characterize vector bundle positivity.

Findings

Schur polynomials in Chern forms of Nakano positive bundles are positive.

Schur polynomials in Griffiths positive bundles are weakly-positive under certain conditions.

Positivity of Schur polynomials does not characterize Griffiths positivity over complex surfaces.

Abstract

We prove that Schur polynomials in Chern forms of Nakano and dual Nakano positive vector bundles are positive as differential forms. Moreover, modulo a statement about the positivity of a "double mixed discriminant" of linear operators on matrices, which preserve the cone of positive definite matrices, we establish that Schur polynomials in Chern forms of Griffiths positive vector bundles are weakly-positive as differential forms. This provides differential-geometric versions of Fulton-Lazarsfeld inequalities for ample vector bundles.An interpretation of positivity conditions for vector bundles through operator theory is in the core of our approach. Another important step in our proof is to establish a certain pushforward identity for characteristic forms, refining the determinantal formula of Kempf-Laksov for homolorphic vector bundles on the level of differential forms. In the same vein, we establish a local version of Jacobi-Trudi identity.Then we study the inverse problem and show that already for vector bundles over complex surfaces, one cannot characterize Griffiths positivity (and even ampleness) through the positivity of Schur polynomials, even if one takes into consideration all quotients of a vector bundle.