Differential operators on a Riemann surface with projective structure (Q1781424)

The author considers arbitrary Riemann surfaces (not necessarily compact or of finite type) equipped with a projective structure and a theta characteristic, i.e. a line bundle which is a square root of the canonical bundle. The complement of the zero section in the total space of the line bundle has a natural symplectic structure, and using the projective structure, this symplectic structure has a canonical quantization by a star product (i.e. a deformation quantization). Using the quantization (via the symbol map) a canonical isomorphism between the space of holomorphic differential operators on certain tensor powers of the theta characteristics \(L\) and the space of holomorphic sections over the Riemann surface \(X\) of certain tensor powers of \(L\) and the canonical bundle \(\Omega\) is constructed. In detail: \[ H^0(X, \text{Diff}_X^k(L^{\otimes j},L^{\otimes (i+j+2k)})) \cong \bigoplus_{l=0}^{k} H^0(X, L^{\otimes i} \otimes \Omega_X^{\otimes l}), \] for \(i,j\in\mathbb Z, n\geq 0\) and \(i\notin [-2(k-1),0]\).