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 ${\rm SO}(3,{\mathbb C})$--operssingularat$S$. 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 $J^2((TX)\otimes {\mathcal O}_X(S))$singularover$S$, satisfying certain natural geometric conditions.