  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 $\Bbb C^n$ 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.
