Integral inequalities for holomorphic maps and applications

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, rigidity and degeneracy theorems without assuming any pointwise curvature signs both the domain target manifolds are proved, in which key roles played by total integration of function first eigenvalue second Ricci an almost nonpositivity notion sectional introduced our previous work. also apply these to discuss infinite-time singularity type Kahler-Ricci flow. The equality case is characterized special settings.