Gelfand-Fuks cohomology of vector fields on algebraic varieties

TL;DR

This work develops an algebraic version of Gelfand–Fuks cohomology for vector fields on affine varieties by using differentiable $AV$-modules and jets. A key advance is the jet-based reduction to the Lie algebra of polynomial jets, enabling a semi-direct product decomposition $A\widehat{\#}V\cong V\ltimes(A\widehat{\otimes}\mathcal{L}_+)$ and a Künneth-type formula for cohomology, which yields a clean factorization $H_{GF}^*(V,M)\cong H_A^*(V,S)\otimes H^*(\mathcal{L}_+,W)$ for varieties with uniformizing parameters. In the tensor-module case, this specializes to $H_{GF}^*(V,A\otimes W)\cong H^*_{dR}(X)\otimes H^*(\mathcal{L}_+,W)$, linking algebraic GF cohomology to de Rham cohomology and the positive-graded Lie algebra cohomology of $\mathcal{L}_+$. The authors compute explicit cohomology for affine space, the torus, and Krichever–Novikov algebras, providing concrete, finite-dimensional descriptions and broadening the computational reach of Gelfand–Fuks theory in algebraic geometry contexts.

Abstract

For an affine algebraic variety, we introduce algebraic Gelfand-Fuks cohomology of polynomial vector fields with coefficients in differentiable $AV$-modules. Its complex is given by cochains that are differential operators in the sense of Grothendieck. Using the jets of vector fields, we compute this cohomology for varieties with uniformizing parameters. We prove that in this case, Gelfand-Fuks cohomology with coefficients in a tensor module decomposes as a tensor product of the de Rham cohomology of the variety and the cohomology of the Lie algebra of vector fields on affine space, vanishing at the origin. We explicitly compute this cohomology for affine space, the torus, and Krichever-Novikov algebras.

Theorems & Definitions (33)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Lemma 2.1
• proof
• Definition 2.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.1