CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCJM
Abstract view

The algebraic de Rham cohomology of representation varieties

  • Eugene Z. Xia,
    Department of Mathematics, National Cheng Kung University, Tainan 70101
Format:   LaTeX   MathJax   PDF  

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 surface, algebraic group, representation variety, de Rham cohomology
MSC Classifications: 14H10, 13D03, 14F40, 14H24, 14Q10, 14R20 show english descriptions Families, moduli (algebraic)
(Co)homology of commutative rings and algebras (e.g., Hochschild, Andre-Quillen, cyclic, dihedral, etc.)
de Rham cohomology [See also 14C30, 32C35, 32L10]
unknown classification 14H24
Surfaces, hypersurfaces
Group actions on affine varieties [See also 13A50, 14L30]
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 14H24
14Q10 - Surfaces, hypersurfaces
14R20 - Group actions on affine varieties [See also 13A50, 14L30]
 

© Canadian Mathematical Society, 2017 : https://cms.math.ca/