Singular holomorphic Morse inequalities
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 technical. First, one ``simplifies'' the situation: by means of monoidal transforms with smooth centers, one obtains a manifold \(X'\), where the induced Nadel ideal defines a normal crossings divisor. Then one applies known ``Morse inequalities'' in this new situation. Finally, one compares the cohomologies of \(X\) and \(X'\), which requires several lemmas and the use of a Leray spectral sequence. In the final section of the paper (independent of the rest) a conjecture is proposed: if \(X\) is a compact analytic manifold, \(E\) and \(F\) are vector bundles over \(X\) of ranks 1 and \(r\) respectively, then \(\dim H^q(X, E^k\otimes F)= r\dim H^q(X, E^k)+ o(k^n)\). The author proves the conjecture in certain cases (if \(X\) is Moishezon or a surface, or if \(X\) is arbitrary and \(q=0\) or \(q= n\)). In spite of its technical nature, the paper is well written and pleasant to read.
scientific article; zbMATH DE number 1463783