Differential operators on complex manifolds with a flat projective structure (Q1282442)

This paper deals with differential operators on complex manifolds equipped with a flat projective structure. These operators generalize the classical Schwarzian derivative on Riemann surfaces. The paper begins with general theory for complex manifolds and later specializes to complex dimension one. Given a flat projective structure on a complex manifold \(X\) of complex dimension \(d\), there exists a holomorphic line bundle \({\mathcal L}\) whose \(d+1\)-power is the canonical line bundle. The author identifies the the jet bundle \(J^n({\mathcal L}^{-n})\) with the \(n\)-th symmetric power of the dual \({\mathcal V}^*\) of the flat \({\mathbb C}^{d+1}\)-bundle \({\mathcal V}\) associated with the flat structure on \(X\). Furthermore, a differential operator of order \(n+1\) \[ \Gamma({\mathcal L}^{-n}) \rightarrow \Gamma(S^{n+1} (\Omega^1_X)\otimes {\mathcal L}^{-n}) \] is obtained, whose symbol identifies with the identity map of \(S^{n+1} (\Omega^1_X)\). This operator can be thought of as a higher dimensional generalization of the Schwarzian derivative and this connection is explored in the paper. The classical Schwarzian derivative \(\mathcal{SD}\) is put in the present context as the operator which associates to the section \((dz)^{-1/2}f\) of \(L^*\) the section \((dz)^{3/2}\mathcal{SD}(f)\) of \(K_X^2\otimes L^*\). Specializing to dimension \(d=1\), the space of differential operators of order \(n\) from \({\mathcal L}^k\) to \({\mathcal L}^l\) is decomposed in terms of spaces of holomorphic sections of powers of \({\mathcal L}\). This result has the corollary that the space of differential operators of order \(n\) from \({\mathcal L}^{-n}\) to \({\mathcal L}^{n+2}\) decomposes as the direct sum of \(H^0(X,K_X^i)\) where \(K_X^i\) is the \(i\)th power of the canonical line bundle; \(i\) ranges over \(0,\dots,n+1\). A similar result is proved for differential operators between possibly different line bundles over \(X\).