On the numerically effective compact Kähler manifolds of Ricci class (Q1384229)

The author proves that, if \(X\) is a compact Kähler manifold of dimension \(n\), with the bundle \(K_X^{-1}\) numerically effective, then \(h^1(X,{\mathcal O}_X)\leq n\). Moreover, if \(\Gamma \) is a subgroup of the fundamental group of \(X\) generated by a finite number of elements, they prove that then there exists its normal subgroup \(\Gamma _1\) of finite index generated by at most \(4^{2n}+1\) elements. Here, the bundle \(L\) is numerically effective one if for each \(\varepsilon >0\) there exists a metric \(h_{\varepsilon }\) on \(L\) such that \(\Theta _{h_{\varepsilon }}(L)\geq -\varepsilon \omega\), where \(\omega \) is a fixed Hermitian metric on \(X\) and \(L\rightarrow X\) is a holomorphic bundle.