Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann

TL;DR

This paper explores the connections between Monge-Ampère integrals in holomorphic Morse inequalities and asymptotic cohomology estimates for line bundle tensor powers, inspired by Riemann's foundational work.

Contribution

It proposes potential relationships between complex geometric integrals and cohomology estimates, extending classical Riemann-Roch ideas in a modern algebraic geometric context.

Findings

Identifies links between Monge-Ampère integrals and cohomology asymptotics

Proposes conjectural general statements in complex geometry

Builds on Riemann's foundational concepts in a new setting

Abstract

The goal of this note is to present the potential relationships between certain Monge-Amp\`ere 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 are still conjectural and 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.