On the Albanese map of compact Kähler manifolds with numerically effective Ricci curvature (Q5929596)

The author gives a differential-geometric approach to the study of a compact Kähler manifold \(X\) with numerically effective Ricci class whose Albanese map is surjective. So, the author proves that the Albanese map of \(X\) is surjective, assuming that \((X,\omega)\) is a compact complex manifold, \((\omega _k)_k\) is a sequence of Kähler metrics on \(X\) such that: NEWLINENEWLINENEWLINE(i) for each \(k>0\), the metric \(\omega _k\) belongs to the cohomology class \(\{ \omega \} \), NEWLINENEWLINENEWLINE(ii) the Ricci curvature of \(\omega _k\) is bounded from below by \(-1/k\), NEWLINENEWLINENEWLINE(iii) the diameters \(d_k:=\) diam\( (X,\omega _k)\) satisfy \(d_k/\sqrt{k}\to 0\) as \(k\) goes to infinity. NEWLINENEWLINENEWLINEThe author also points out that a compact Kähler manifold, whose Ricci class is numerically effective and integrable, possesses a sequence of metrics with the previously mentioned properties (i)--(iii). In the final part of this paper some results are given in the case that the sequence of diameters \((d_k)_k\) goes to infinity.