K\"ahler-Einstein cone metrics and parabolic bundles

Abstract

Let $X$ be a compact K\"ahler manifold, let $D=\sum_iD_i$ be a simple normal crossing divisor, and fix $\beta_i\in(0,1)$. We introduce a parabolic structure $TX_*$ on $TX$, given along each $D_i$ by the filtration $0\subset TD_i\subset TX|_{D_i}$ with weight $w_i=1-\beta_i$ on the normal quotient. If $X\setminus D$ carries a K\"ahler--Einstein metric $\omega$ with cone angle $2\pi\beta_i$ along $D_i$, we show that $TX_*$ 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.