We construct Kähler–Einstein metrics with negative scalar curvature near an isolated log canonical (non-log terminal) singularity.Such metrics are complete near the singularity if the underlying space has complex dimension 2. We also establish a stability result for Kähler–Einstein metrics near certain types of isolated log canonical singularity. As application, for complex dimension 2 log canonical singularity, we show that any complete Kähler–Einstein metric of negative scalar curvature near an isolated log canonical (non-log terminal) singularity is smoothly asymptotically close to model Kähler–Einstein metrics from hyperbolic geometry.
Complete Kähler–Einstein Metric on Stein Manifolds With Negative Curvature
Abstract We show the existence of complete negative Kähler–Einstein metric on Stein manifolds with holomorphic sectional curvature bounded from above by a negative constant. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler–Einstein metric using the normalized Kähler–Ricci flow.
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- Award ID(s):
- 1709894
- NSF-PAR ID:
- 10258182
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Every compact Kähler manifold with negative first Chern class admits a unique metric such that . Understanding how families of these metrics degenerate gives insight into their geometry and is important for understanding the compactification of the moduli space of negative Kähler–Einstein metrics. I study a special class of such families in complex dimension two. Following the work of Sun and Zhang in the Calabi–Yau case [2019, arXiv:1906.03368], I construct a Kähler–Einstein neck region interpolating between canonical metrics on components of the central fiber. This provides a model for the limiting geometry of metrics in the family.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnaa142
