The Algebraic de Rham Cohomology of Representation Varieties

Xia, Eugene Z.

Canadian journal of mathematics, Tome 70 (2018) no. 3, pp. 702-720

Résumé

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.

Détail

DOI : 10.4153/CJM-2017-010-8

Mots-clés : 14H10, 13D03, 14F40, 14H24, 14Q10, 14R20, surface, algebraic group, representation variety, de Rham cohomology

Bibliographie

[1] [1] Brieskorn, E., Die Monodromie der isolierten Singularitäten von Hyperflächen. Manuscripta Math. 2(1970), 103–161. Google Scholar | DOI

[2] [2] Cox, D., Little, J., and O'Shea, D., Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Third ed., Undergraduate Texts in Mathematics, Springer, New York, 2007. http://dx.doi.Org/10.1007/978-0-387-35651-8 Google Scholar

[3] [3] Decker, W., Greuel, G.-M., Pfister, G., and Schönemann, H., Singular 4-0-2—A computer algebra system for polynomial computations, http://www.singular.uni-kl.de Google Scholar

[4] [4] Deligne, P., Équations différentielles á points singuliers réguliers. Lecture Notes in Mathematics, 163, Springer-Verlag, Berlin-New York, 1970. Google Scholar

[5] [5] Deligne, P. and Katz, N., eds., Groupes de monodromie en géométrie algébrique. II. In: Séminaire de Géométrie Algébrique du Bois-Marie 1967-1969 (SGA 7II). Lecture Notes in Mathematics, 340, Springer-Verlag, Berlin-New York, 1973. Google Scholar

[6] [6] Eisenbud, D., Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995. Google Scholar | DOI

[7] [7] Goldman, W. M., Trace coordinates on Fricke spaces of some simple hyperbolic surfaces. In: Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc, Zürich, 2009, pp. 611–684. Google Scholar | DOI

[8] [8] Gelfand, S. I. and Manin, Y., Methods of homological algebra. Second ed., Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. Google Scholar | DOI

[9] [9] Goldman, W. M. and Neumann, W. D., Homological action of the modular group on some cubic moduli spaces. Math. Res. Lett. 12(2005), no. 4, 575–591. Google Scholar | DOI

[10] [10] Grayson, D. and Stillman, M.. Macaulay 2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/ Google Scholar

[11] [11] Grothendieck, A., On the de Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 29(1966), 95–103. Google Scholar

[12] [12] Grothendieck, A., Sur quelques points d'algèbre homologique. Tôhoku Math. J. (2) 9(1957), 119–221. Google Scholar

[13] [13] Hartshorne, R., On theDe Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. 45(1975), 5–99. Google Scholar

[14] [14] Katz, N. M., Rigid local systems. Annals of Mathematics Studies, 139, Princeton University Press, Princeton, NJ, 1996. Google Scholar | DOI

[15] [15] Katz, N. M. and Oda, T., On the differentiation of de Rham cohomology classes with respect to parameters. J. Math. Kyoto Univ. 8(1968), 199–213. http://dx.doi.Org/10.1215/kjm/1250524135 Google Scholar

[16] [16] Malgrange, B., Sur les points singuliers des équations différentielles. Enseignement Math. (2) 20 1974), 147–176. Google Scholar

[17] [17] Milnor, J., Singular points of complex hypersurfaces. Annals of Mathematics Studies, 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Google Scholar

[18] [18] Oaku, T. and Takayama, N., An algorithm for de Rham cohomology groups of the complement of an affme variety via D-module computation. In: Effective methods in algebraic geometry (Saint-Malo, 1998). J. Pure Appl. Algebra 139(1999), no. 1-3, 201–233. http://dx.doi.Org/10.1016/S0022-4049(99)00012-2 Google Scholar

[19] [19] Scheiblechner, P., Effective de Rham cohomology-the hypersurface case. In: ISSAC 2012-Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2012, pp. 305–310. Google Scholar | DOI

[20] [20] Scheiblechner, P., Castelnuovo-Mumford regularity and computing the de Rham cohomology of smooth protective varieties. Found. Comput. Math. 12(2012), no. 5, 541–571. http://dx.doi.Org/10.1007/s10208-012-9123-y Google Scholar

[21] [21] Schulze, M., Algorithms for the Gauss-Manin connection. J. Symbolic Comput. 32(2001), no. 5, 549–564. http://dx.doi.Org/10.1006/jsco.2001.0482 Google Scholar

[22] [22] Walther, U., Algorithmic determination of the rational cohomology of complex varieties via differential forms. In: Symbolic computation: solving equations in algebra, geometry, and engineering (South Hadley, MA, 2000), Contemp. Math., 286, American Mathematical Society, Providence, RI, 2001, pp. 185–206. http://dx.doi.Org/10.1090/conm/286/04763 Google Scholar

[23] [23] Weibel, C. A., An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. http://dx.doi.Org/10.1017/CBO9781139644136 Google Scholar

[24] [24]Wolfram Research, Inc., Mathematica. Version 7.0. Champaign, IL, 2008. Google Scholar