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

Nariya Kawazumi
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Abstract

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.

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Nariya Kawazumi. 2005-08-21. An application of the second Riemann continuation theorem to cohomology of the Lie algebra of vector fields on the complex line. https://arxiv.org/abs/math/0508390

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