A vanishing theorem for semipositive line bundles over non-Kähler manifolds. (English) Zbl 0577.32031

Siu, Yum-Tong

J. Differ. Geom. 19, 431-452 (1984).

The main result is the following. Let M be a compact complex manifold and L be a Hermitian holomorphic line bundle on M. Suppose the curvature form of L being everywhere semipositive and being strictly positive outside a set of measure zero. Then \(H^ q(M,LK_ M)=0\) for \(q\geq 1\), where \(K_ M\) is the canonical line bundle of M.
A characterization of Moišezon manifolds is given. Connections with the conjecture of Grauert-Riemenschneider and the eigenvalue conjecture are discussed, too.

Reviewer: A.Pankov

MSC:

32L20	Vanishing theorems
32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
32J99	Compact analytic spaces
32J15	Compact complex surfaces

Keywords:

compact complex manifold; Hermitian holomorphic line bundle; curvature form; Moišezon manifolds; conjecture of Grauert-Riemenschneider; eigenvalue conjecture