Branched projective structures, branched SO(3,C)-opers and logarithmic connections on jet bundle

We study the branched holomorphic projective structures on a compact Riemann surface $X$ with a fixed branching divisor $S, =, sum_{i=1}^d x_i$, where $x_i ,in, X$ are distinct points. After defining branched ${rm SO}(3,{mathbb C})$--opers, we show that the branched holomorphic projective structures on $X$ are in a natural bijection with the branched ${rm SO}(3,{mathbb C})$--opers singular at $S$. It is deduced that the branched holomorphic projective structures on $X$ are also identified with a subset of the space of all logarithmic connections on $J^2((TX)otimes {mathcal O}_X(S))$ singular over $S$, satisfying certain natural geometric conditions.