An application of the second Riemann continuation theorem to cohomology of the Lie algebra of vector fields on the complex line

We study cohomology groups of the Lie algebra of vector fields on the complex line, $W_1$, with values in the tensor fields in several variables. From a generalization by Scheja of the second Riemann (Hartogs) continuation theorem, we deduce a cohomology exact sequence of the subalgebra of $W_1$ consisting of vectors having a zero at the origin. As applications, we compute the cohomology algebra of $W_1$ with values in the functions on ${\mathbb{C}}^n$ explicitly, and establish a certain vanishing theorem for the cohomology of $W_1$ with values in the quadratic differentials in several variables, which is closely related to the moduli space of Riemann surfaces.