Singular holomorphic Morse inequalities
type: Item
instance of: scholarly article
title: Singular holomorphic Morse inequalities
zbMATH Open document ID: 0966.32011
DOI: 10.1007/BF02921793
author: Laurent Bonavero
published in: The Journal of Geometric Analysis
publication date: 2000-06-22T00:00:00Z
review text: The main result of this paper is the following theorem. Let \(X\) be a compact complex manifold of dimension \(n\), \(E\) a holomorphic line bundle over \(X\), equipped with a metric \(h\) (which might be singular). It is known that to the pair \((E,h)\) and an integer \(k>0\) one may associate a sheaf of ideals \({\mathcal I}_k(h)\subset{\mathcal O}_X\), the so-called ``Nadel's multiplier sheaf''. Let \(\Theta(E)\) be the curvature current of \(E\), \(X(q,E)\) the analytic open subset of \(X\) of points where the metric is smooth, with index \(\leq q\), where \(0\leq q\leq n\). Finally, let \(F\) be a holomorphic vector bundle on \(X\), of rank \(r\). Then, the author proves the following ``Morse's inequality'': \[ \sum^q_{j=0} \dim H^j(X,{\mathcal O}(E^k\otimes F)\otimes{\mathcal I}_k(h))\leq r(k^n/n!) \int_{X(q,E)} (-1)^q \Theta(E)^n+ o(k^n).\tag{\(*\)} \] The main motivation for this formula is to investigate the conjecture of Grauert and Riemenschneider about the characterization of Moishezon manifolds (i.e., the compact ones having \(n\) algebraically independent meromorphic functions, where \(n\) is the dimension) as those admitting a suitable line bundle with semi-positive, generically positive curvature. This would be an analog of a celebrated theorem of Kodaira characterizing projective manifolds as those admitting a positive line bundle. Y. T. Siu and (independently) J. P. Demailly gave sufficient conditions for a manifold to be Moishezon, in terms of the existence of an appropriate line bundle with a smooth metric. In his work, Demailly used a certain type of Morse inequalities (which are generalized in the present paper). With formula \((*)\), the author gives a characterization of Moishezon manifolds, in terms of the existence of a current \(T\) on \(X\) satisfying suitable conditions. He also expands and simplifies previous work by J. Kollar, showing that the conditions of Siu-Demailly are not necessary. The proof of formula \((*)\) is rather technica…
reviewed by: Augusto Nobile
Mathematics Subject Classification ID: 32J18
Mathematics Subject Classification ID: 58J99
Mathematics Subject Classification ID: 32J99
Mathematics Subject Classification ID: 32L99
zbMATH DE Number: 1463783
zbMATH Keywords: vector bundle
zbMATH Keywords: curvature
zbMATH Keywords: complex manifold
zbMATH Keywords: holomorphic line bundle
zbMATH Keywords: Moishezon manifolds
zbMATH Keywords: current
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1463783