Volume growth of Kähler–Einstein metric over quasi-projective manifolds with boundary of maximal or minimal Kodaira dimension

Abstract

In this paper, we make some progress about a boundary behavior of the almost-complete Kähler–Einstein metric of negative Ricci curvature on a quasi-projective manifold with semiample log-canonical bundle. First its volume growth near the boundary is investigated in terms of the Kodaira dimension of the boundary, and then we characterize the boundary to be of general type via the volume growth. Moreover the volume growth is determined in the case of a Calabi–Yau boundary. We also affirmatively solve a modified version of the conjecture suggested previously by the author about the residue of the Kähler–Einstein metric if the boundary is a smooth finite quotient of an abelian variety.