Holomorphic Morse Inequalities and Asymptotic Cohomology Groups: A Tribute to Bernhard Riemann

Abstract

The goal of this note is to present the potential relationships between certain Monge-Ampere integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of line bundles, as recently introduced by algebraic geometers. The expected most general statements, which are still conjectural, certainly owe a debt to Riemann’s pioneering work, which led to the concept of Hilbert polynomials and to the Hirzebruch-Riemann-Roch formula during the XX-th century.