Bidifferentials, Lagrangian projections and the Virasoro extension

Let $C$ be a smooth projective curve over an algebraically closed field $k$ of characteristic zero. We prove that a Lagrangian supplement of $H^0(C, \Omega_C)$ in the de Rham cohomology group $H^1_{dR}(C)$ determines and is determined by a particular type of symmetric bidifferential on $C^2$ (its polar divisor must be twice the diagonal and have biresidue one along it). When $k$ is the complex field, a natural choice of such supplement is $H^{0,1}(C)$ and we show that this corresponds with the bidifferential that after a twist is the rational $2$-form on $C^2$ found by Biswas-Colombo-Frediani-Pirola. We determine the cohomology class carried by that $2$-form and define an analogue of this form as rational $n$-form on $C^n$ that is regular on the $n$-point configuration space of $C$. The proof relies on a local version of the above correspondence, which can be stated in terms of a complete discrete valuation ring. We use this local version also to construct in a natural manner the Virasoro extension of the Lie algebra of derivations of a local field.