The limits on boundary of orbifold Kähler–Einstein metrics and Kähler–Ricci flows over quasi-projective manifolds

Abstract

In this paper, we consider a sequence of orbifold Kahler–Einstein metrics \((\omega _{X_m})_m\) and normalized Kahler–Ricci flows \((\omega _{X_m}(t))_m\) on a quasi-projective manifold \(X=\overline{X} {\setminus } \overline{D}\) for a projective manifold \(\overline{X}\) and a divisor \(\overline{D}\) of \(\overline{X}\) with simple normal crossings such that \(K_{\overline{X}}+\overline{D}\) is ample. For sufficiently large \(m, \omega _{X_m}\) or \(\omega _{X_m}(t)\) is the orbifold Kahler–Einstein metric or orbifold normalized Kahler–Ricci flow on \(X\) equipped with the orbifold structure with respect to the divisor \(1/m\overline{D}\), respectively. The main theorem of this paper is that the limit on the boundary \(\overline{D}\) or the inside \(X\) of the sequence \((\omega _{X_m}(t))_m\) as \(m \rightarrow \infty \) is exactly the complete normalized Kahler–Ricci flow on \(\overline{D}\) or \(X\), respectively. Moreover our method of the proof also leads to a result that on the boundary \(\overline{D}\) the sequence \((\omega _{X_m})_m\) converges to the complete Kahler–Einstein metric on \(\overline{D}\) as \(m \rightarrow \infty \).