Kähler-Einstein cone metrics and parabolic bundles

Let $X$ be a compact Kähler manifold, let $D=\sum_iD_i$ be a simple normal crossing divisor, and fix $β_i\in(0,1)$. 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 $w_i=1-β_i$ on the normal quotient. If $X\setmi