Kähler-Einstein cone metrics and parabolic bundles

Abstract

Let $X$ be a compact Kähler manifold, let $D={\sum }_i{D}_i$ be a simple normal crossing divisor, and fix ${\beta }_i\in (0,1)$. We introduce a parabolic structure $T{X}_{\ast }$ on $TX$, given along each $D_i$ by the filtration $0\subset T{D}_i\subset TX{|}_{D_i}$ with weight $w_i=1-{\beta }_i$ on the normal quotient. If $X\setminus D$ carries a Kähler--Einstein metric $\omega$ with cone angle $2\pi {\beta }_i$ along $D_i$, we show that $T{X}_{\ast }$ is parabolic polystable with respect to $[\omega ]$. When the Einstein constant $\lambda$ is non-zero, we introduce parabolic structures on the Tian extension ($\lambda >0$) and on Simpson's canonical Higgs bundle ($\lambda <0$), 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.

arXiv comment22 pages