Boundary behavior of Kähler-Einstein metric of negative Ricci curvature over quasi-projective manifolds with boundary of general type

Abstract

In this paper, we discuss an asymptotic boundary behavior of the complete Kähler-Einstein metric of negative Ricci curvature on a quasi-projective manifold with semiample log-canonical bundle. In particular, we focus our attention on its relations with degeneration of positivity for the log-canonical bundle on the boundary divisor. Based on a pioneering result due to G. Schumacher, a fundamental conjecture about the relations is proposed in this paper. Moreover it is also proved that the conjecture actually holds in the case when the boundary divisor is of general type.