The Algebraic de Rham Cohomology of Representation Varieties
Canadian journal of mathematics, Tome 70 (2018) no. 3, pp. 702-720
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The $\text{SL}\left( 2,\mathbb{C} \right)$ -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 $\text{SL}\left( 2,\mathbb{C} \right)$ -representation varieties of the one-holed torus.
Mots-clés :
14H10, 13D03, 14F40, 14H24, 14Q10, 14R20, surface, algebraic group, representation variety, de Rham cohomology
Xia, Eugene Z. The Algebraic de Rham Cohomology of Representation Varieties. Canadian journal of mathematics, Tome 70 (2018) no. 3, pp. 702-720. doi: 10.4153/CJM-2017-010-8
@article{10_4153_CJM_2017_010_8,
author = {Xia, Eugene Z.},
title = {The {Algebraic} de {Rham} {Cohomology} of {Representation} {Varieties}},
journal = {Canadian journal of mathematics},
pages = {702--720},
year = {2018},
volume = {70},
number = {3},
doi = {10.4153/CJM-2017-010-8},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2017-010-8/}
}
TY - JOUR AU - Xia, Eugene Z. TI - The Algebraic de Rham Cohomology of Representation Varieties JO - Canadian journal of mathematics PY - 2018 SP - 702 EP - 720 VL - 70 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2017-010-8/ DO - 10.4153/CJM-2017-010-8 ID - 10_4153_CJM_2017_010_8 ER -
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