Characterization of positive Chern-form polynomials

Characterize the polynomials in Chern classes or Chern forms that define positive cohomology classes or differential forms, in an appropriate sense, for Griffiths-positive holomorphic vector bundles.

Background

Griffiths asked which polynomials in the Chern classes or Chern forms of a Griffiths-positive vector bundle are positive. The class-level version was completely characterized by Fulton and Lazarsfeld: the positive polynomials are precisely positive combinations of Schur polynomials. The corresponding differential-form-level problem is substantially more difficult and remains largely unresolved.

References

At the form level, Griffiths' above question is more subtle and in its general form is largely open.

— Chern numbers on positive vector bundles and combinatorics  (2501.08833 - Li, 15 Jan 2025) in Section 1, Introduction