The aim of this research project is to investigate the positivity of certain families of differential forms arising from Griffiths positive hermitian holomorphic vector bundles.
On the cohomology classes level the issue has been completely solved by Fulton and Lazarsfeld, who showed that the positive polynomials in the Chern classes of an ample vector bundle are exactly those belonging to the positive convex cone spanned by the Schur polynomials in the Chern classes of the bundle.
More generally, for differential forms the question remains open in both directions.
Partial answers are due to the works of Guler and Diverio, which show that Segre forms (namely, special Schur polynomials in the Chern forms) are positive for Griffiths positive vector bundles.
Motivated by these results, this research aims to investigate which differential forms in the positive convex cone spanned by the Schur polynomials in the Chern forms are positive for Griffiths positive vector bundles.
Positivity in complex geometry is a very fertile field of research. As the previous literature teaches us, concrete information regarding the geometry of the base manifold of a holomorphic vector bundle can be deduced from the positivity of the bundle itself. Two crucial results in this area are the Kodaira Embedding Theorem, which establishes the equivalence between the algebraicity of a compact complex manifold and the existence of a positive line bundle on that manifold, and the Nakai criterion, which completely characterizes the positivity of a line bundle over a proper scheme precisely in terms of the positivity of certain Chern numbers.
Given that in rank 1 positivity allows us to deduce important geometric results, it is natural to investigate vector bundles of higher rank. Griffiths started a program in [Gr69] with the aim of studying the relationships among the main notions of positivity for vector bundles, including ampleness and Griffiths positivity. This program is particularly interesting since we cannot have results analogous to the Nakai Criterion in rank greater than 1. The results obtained so far as well as their applications in complex birational geometry and in the Minimal Model Program led to a great expansion of knowledge about the geometry of complex manifolds. Hence, it is essential to have precise translations between algebraic positivity and its counterpart in a differential geometric sense. Such a translation is missing for Fulton-Lazarsfeld's Theorem and obtaining it would be a basic achievement. Such a result will allow a better understanding of the metric aspects of the manifolds underlying a Griffiths positive bundle. An immediate consequence, among others, is that the positivity of the aforementioned forms will produce new inequalities between differential forms, which impose strong geometric constraints on the base manifold.