canonical metrics or measures in complex geometry and positivity of canonical bundle

2015 Fiscal Year Final Research Report

• Project/Area Number: 26887036
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Kogakuin University (2015); Tokyo Denki University (2014)
• Principal Investigator: Kikuta Shin 工学院大学, 公私立大学の部局等, 准教授 (40736790)
• Project Period (FY): 2014-08-29 – 2016-03-31
• Keywords: 完備ケーラー・アインシュタイン計量の境界挙動 / 準射影代数多様体 / 境界における対数的標準束の正値性の退化 / 特異ケーラー・アインシュタイン計量 / 複素双曲多様体のトロイダルコンパクト化 / 体積増大度

Outline of Final Research Achievements

In this project, we studied a boundary behavior of the complete Kahler-Einstein metric of negative Ricci curvature over quasi-projective manifolds. In particular, a case when positivity of the log-canonical bundle is strictly degenerate on the boundary was mainly discussed. We first proved that when the boundary is of general type, the metric asymptotically approaches to the singular Kahler-Einstein metric in the directions tangent to the boundary. This convergence follows on the whole of the boundary as a current, and on the ample locus of the canonical bundle in the smooth topology. Moreover it was also established that for any troidal compactifications of complex hyperbolic manifolds, the metric asymptotically and smoothly approaches to 0 in the tangent directions.

Free Research Field

複素幾何学