On Hodge-Riemann Cohomology Classes

Abstract

We prove that Schur classes of nef vector bundles are limits of classes that have a property analogous to the Hodge-Riemann bilinear relations. We give a number of applications, including (1) new log-concavity statements about characteristic classes of nef vector bundles (2) log-concavity statements about Schur and related polynomials (3) another proof that normalized Schur polynomials are Lorentzian.

Hodge-Riemann Relations for Schur Classes in the Linear and Kähler Cases

“…The linear algebra machinery we develop in this paper is an abstraction of the arguments in [RT19]. In fact, combining what is written here with [RT21] reproves the main results of [RT19].…”

“…The proof we give of our main result will depend on our previous work on Schur classes of vector bundles. Here we state and sketch the proofs of two results from [RT19] and [RT21] which will be used in an essential way in Proposition 4.7. (In fact we will use slight generalizations that allow the base space to be irreducible rather than smooth.)…”

“…Comparison with previous work: Our first work in this subject is [RT19] in which we prove the Hodge-Riemann property for Schur classes of ample bundles. We continue this in [RT21] in which we emphasize more the importance of the weak Hodge-Riemann property (which is much easier to prove) and from this we develop various inequalities among characteristic classes of nef vector bundles. The linear algebra machinery we develop in this paper is an abstraction of the arguments in [RT19].…”