Vanishing theorems on complete Kähler manifolds and their applications
In this paper vanishing theorems for \(L^2\)-sections of the holomorphic vector bundles, \(E\), over complete Kähler manifolds, \(M\), are studied. Also, some applications are studied. In the first part, for example, it is shown that if \(M\) is a non-parabolic Kähler manifold with nonnegative Ricci curvature and \(E\) is a quasi-positive line bundle over \(M\), then the degree of \(E\) has to be infinity. Moreover, motivated by the work of \textit{Y.-T. Siu} and \textit{S.-T. Yau} [Ann. Math., II. Ser. 105, 225-264 (1977; Zbl 0358.32006) and \textit{R. E. Greene} and \textit{H. Wu} [Duke. Math. J. 49, 731-756 (1982; Zbl 0513.53045)] the gap phenomenon is established: Suppose \(M^m\) is a complete non-parabolic Kähler manifold with nonnegative Ricci curvature. If the Ricci curvature is quasi-positive then the total scalar curvature \(\int_MS(x) dv=\infty\). The estimate of the dimension of the space of \(L^2\) holomorphic sections of a line bundle under certain conditions is obtained as a generalization of a result of \textit{N. Mok} [Math. Z. 191, 303-317 (1986; Zbl 0585.32032)]. Huber's theorem is also generalized. The second part of the paper is related to the following conjecture of Yau and Green-Wu. Conjecture. Suppose \(M\) is a complete noncompact Kähler manifold with positive holomorphic bisectional curvature. Then \(M\) is biholomorphic to \(\mathbb{C}^n\). \textit{N. Mok, Y.-T. Siu} and \textit{S.-T. Yau} proved the conjecture under the additional assumptions [Compos. Math 44, 183-218 (1984; Zbl 0531.32007)]. The improved version of the Mok-Siu-Yau's theorem is proved in the paper.
scientific article; zbMATH DE number 1367998