Positivity of the third Chern form for Griffiths positive vector bundles

TL;DR

This paper proves the positivity of the third Chern form for Griffiths positive vector bundles of rank three over complex threefolds, resolving a long-standing question by Griffiths from 1969.

Contribution

It establishes the weak positivity of the third Chern form and all Schur forms for rank three Griffiths positive bundles, settling a 54-year-old open problem.

Findings

Proves positivity of the double mixed discriminant for positive linear maps.

Shows weak positivity of the third Chern form for Griffiths positive bundles.

Confirms all Schur forms are weakly positive for rank three bundles over threefolds.

Abstract

In this paper, we prove the positivity of the double mixed discriminant associated with a positive linear map between spaces of third-order complex matrices, thereby settling the three-dimensional case of Finski's open problem. As an application, we obtain the weak positivity of the third Chern form for Griffiths positive vector bundles. Moreover, we show that all Schur forms are weakly positive for Griffiths positive vector bundles of rank three over complex threefolds. This yields a complete affirmative answer, in the case where both the rank and the dimension are three, to the question posed by Griffiths in 1969.