A criterion for Lie algebroid connections on a compact Riemann surface (Q6584169)

Given a complex manifold \(M\) and a locally free sheaf \(V\) on \(M\) (or more generally a holomorphic principal bundle) its \textit{Atiyah bundle} is an example of a \textit{Lie algebroid}.\N\NIn general such a structure is defined with respect to a holomorphic homomorphism \(\phi: V \to TM\), the so-called anchor map. A Lie algebroid \((V,\phi)\) is a Lie algebra structure \( [ - , - ] : V \otimes_{\mathbb C} V \to V \) that satisfies the Leibniz rule \([f s. t] = f [s,t]- \phi(t)(f)s\) for arbitrary local holomorphic sections \(s,t\) of \(V\) and holomorphic functions \(f\).\N\NFor any Lie algebroid \((V,\phi)\), the notion of a holomorphic connection on a holomorphic vector bundle \(E\) on \(M\) is generalized by using the anchor map: A \textit{Lie algebroid connection} is a first-order holomorphic differential operator \( D: E \to E \otimes V^* \) such that \(D(fs) = fD(s)+ s \otimes \phi^*(df)\) for all (locally defined) holomorphic sections \(s\) of \(E\) holomorphic functions \(f\) on \(M\), where \(\phi^*\) is the dual anchor map. For \(V=TM\) and \(\phi=id\) this gives the definition of a holomorphic connection.\N\NA first result of the article is the characterization of holomorphic connections as splittings of certain exact sequences involving jet differentials. This approach is used to prove the main theorem on the existence of Lie algebroid connections.\N\NBy assumption \((V,\phi)\) is a Lie algebroid with a \textit{stable} underlying holomorphic vector bundle on a Riemann surface \(X\). The main theorem states that every holomorphic vector bundle \(E\) on \(X\) possesses a Lie algebroid connection, if \(\phi\) is not an isomorphism. In case it is, such a connection exists, if and only if the degree of each indecomposable component of \(E\) is zero.