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# The Algebraic de Rham Cohomology of Representation Varieties

Published:2017-06-07
Printed: Jun 2018
• Eugene Z. Xia,
Department of Mathematics, National Cheng Kung University, Tainan 70101
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## Abstract

The $\operatorname{SL}(2,\mathbb C)$-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth $\operatorname{SL}(2,\mathbb C)$-representation varieties of the one-holed torus.
 Keywords: surface, algebraic group, representation variety, de Rham cohomology
 MSC Classifications: 14H10 - Families, moduli (algebraic) 13D03 - (Co)homology of commutative rings and algebras (e.g., Hochschild, Andre-Quillen, cyclic, dihedral, etc.) 14F40 - de Rham cohomology [See also 14C30, 32C35, 32L10] 14H24 - unknown classification 14H2414Q10 - Surfaces, hypersurfaces 14R20 - Group actions on affine varieties [See also 13A50, 14L30]

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