On the classification of three-dimensional compact Kähler manifolds of nonnegative bisectional curvature (Q798968)

Let M be an n-dimensional Kähler manifold. M is said to be of quasipositive Ricci curvature if the Ricci tensor is positive semidefinite everywhere and positive definite somewhere. The main result of this paper is the following interesting classification theorem: Let M be a three-dimensional compact Kähler manifold of nonnegative bisectional curvature. If M has quasipositive Ricci curvature, then M is biholomorphic to one of the following \[ P^ 3({\mathbb{C}}),\quad Q^ 3({\mathbb{C}}),\quad P^ 1({\mathbb{C}})\times P^ 2({\mathbb{C}}),\quad P^ 1({\mathbb{C}})\times P^ 1({\mathbb{C}})\times P^ 1({\mathbb{C}}). \] This theorem is proved by using properties on the solutions of an evolution equation in metric tensors for three-dimensional compact Kähler manifolds. This equation was introduced in the real three-dimensional case by \textit{R. S. Hamilton} [J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)].