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ON HODGE-RIEMANN COHOMOLOGY CLASSES JULIUS ROSS AND MATEI TOMA 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. CINTRODUCTION Since the dawn of time, human beings have asked some fundamental questions: who are we? why are we here? is there life after death? Unable to answer any of these, in this paper we will consider cohomology classes on a compact projective manifold that have a property analogous to the Hard-Lefschetz Theorem and Hodge-Riemann bilinear relations To state our results let X be a projective manifold of dimension d ≥ 2. We say that a co homology class Ω Hd–2,d–2(X; R) has the Hodge-Riemann property if the intersection form QΩ(α, α′c := aΩα′ for α, α′ H 1,1(X ; R) has signature (+, −, −, . .. , −). We write HR(X) = {Ω with the Hodge Riemann property} and HR(.X) for its closure.
