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Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann. (English. French summary) Zbl 1205.32017

Summary: The goal of this note is to present the potential relationships between certain Monge-Ampère 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.

MSC:

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14B05 Singularities in algebraic geometry
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
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