On characteristic forms of positive vector bundles, mixed discriminants and pushforward identities
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 weaklypositive as differential forms. This provides differentialgeometric versions of FultonLazarsfeld 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 KempfLaksov for homolorphic vector bundles on the level of differential forms. In the same vein, we establish a local version of JacobiTrudi 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2020
 arXiv:
 arXiv:2009.13107
 Bibcode:
 2020arXiv200913107F
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Differential Geometry