PreprintarXiv licencearXiv
Kähler-Einstein cone metrics and parabolic bundles
Abstract
Let X be a compact Kähler manifold, let D=∑iDi be a simple normal crossing divisor, and fix βi∈(0,1). We introduce a parabolic structure TX* on TX, given along each Di by the filtration 0⊂ TDi⊂ TX|Di with weight wi=1−βi on the normal quotient. If X∖ D carries a Kähler--Einstein metric ω with cone angle 2πβi along Di, we show that TX* is parabolic polystable with respect to [ω]. When the Einstein constant λ is non-zero, we introduce parabolic structures on the Tian extension (λ>0) and on Simpson's canonical Higgs bundle (λ<0), and prove that these are polystable too. Combined with the Bogomolov--Gieseker inequality for parabolic bundles, this yields a Miyaoka--Yau inequality for these pairs.
Related
- The curvature estimation of the complete Kähler-Einstein metrics on the disk bundles
- Curvature growth and holomorphic splitting for Kähler metrics with cone singularities
- Parabolic Bogomolov--Gieseker Inequality and Pluriharmonic Metrics on Regular Parabolic Higgs Sheaves over Compact Kähler Manifolds
- Instability of conformally Kähler, Einstein metrics
- Local classification of chsc Kähler metrics with cone singularities
BibTeX
@misc{borbon2026kählereinstein,
title = {{Kähler-Einstein cone metrics and parabolic bundles}},
author = {Martin de Borbon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24597},
eprint = {2609.24597},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24597},
}
RIS
TY - UNPB
TI - Kähler-Einstein cone metrics and parabolic bundles
AU - Martin de Borbon
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24597
UR - https://arxiv.org/abs/2609.24597
AB - Let X be a compact Kähler manifold, let D=∑_iD_i be a simple normal crossing divisor, and fix β_i∈(0,1). We introduce a parabolic structure TX_* on TX, given along each D_i by the filtration 0⊂ TD_i⊂ TX|_D_i with weight w_i=1−β_i on the normal quotient. If X∖ D carries a Kähler--Einstein metric ω with cone angle 2πβ_i along D_i, we show that TX_* is parabolic polystable with respect to [ω]. When the Einstein constant λ is non-zero, we introduce parabolic structures on the Tian extension (λ>0) and on Simpson's canonical Higgs bundle (λ<0), and prove that these are polystable too. Combined with the Bogomolov--Gieseker inequality for parabolic bundles, this yields a Miyaoka--Yau inequality for these pairs.
ER -