Algebraic geometry/Topology
An integrable connection on the configuration space of a Riemann surface of positive genus
Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 312-315.

Let X be a Riemann surface of positive genus. Denote by X(n) the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on H1(X(n)) to define an integrable connection on the trivial vector bundle on X(n) with fiber the universal algebra of the Lie algebra associated with the descending central series of π1 of X(n). The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.

Soit X une surface de Riemann de genre positif. Nous notons X(n) l'espace des configurations de n points distincts sur X. Nous utilisons l'isomorphisme de comparaison de Betti–de Rham sur H1(X(n)) pour définir une connexion intégrable sur le fibré vectoriel trivial sur X(n), dont la fibre est l'algèbre universelle de l'algèbre de Lie associée à la série centrale descendante du π1 de X(n). La construction s'inspire du système de Knizhnik–Zamolodchikov en genre zéro ; l'intégrabilité résulte des relations de périodes de Riemann.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.02.003
Eskandari, Payman 1

1 Department of Mathematics, University of Toronto, 40 St. George St., Room 6290, Toronto, Ontario, M5S 2E4, Canada
@article{CRMATH_2018__356_3_312_0,
     author = {Eskandari, Payman},
     title = {An integrable connection on the configuration space of a {Riemann} surface of positive genus},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {312--315},
     publisher = {Elsevier},
     volume = {356},
     number = {3},
     year = {2018},
     doi = {10.1016/j.crma.2018.02.003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.02.003/}
}
TY  - JOUR
AU  - Eskandari, Payman
TI  - An integrable connection on the configuration space of a Riemann surface of positive genus
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 312
EP  - 315
VL  - 356
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.02.003/
DO  - 10.1016/j.crma.2018.02.003
LA  - en
ID  - CRMATH_2018__356_3_312_0
ER  - 
%0 Journal Article
%A Eskandari, Payman
%T An integrable connection on the configuration space of a Riemann surface of positive genus
%J Comptes Rendus. Mathématique
%D 2018
%P 312-315
%V 356
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.02.003/
%R 10.1016/j.crma.2018.02.003
%G en
%F CRMATH_2018__356_3_312_0
Eskandari, Payman. An integrable connection on the configuration space of a Riemann surface of positive genus. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 312-315. doi : 10.1016/j.crma.2018.02.003. http://www.numdam.org/articles/10.1016/j.crma.2018.02.003/

[1] Bellingeri, P. On presentations of surface braid groups, J. Algebra, Volume 274 (2004), pp. 543-563

[2] Bellingeri, P.; Gervais, S.; Guaschi, J. Lower central series for Artin Tits and surface braid groups, J. Algebra, Volume 319 (2008), pp. 1409-1427

[3] Bezrukavnikov, R. Koszul DG-algebras arising from configuration spaces, Geom. Funct. Anal., Volume 4 (1994) no. 2, pp. 119-135

[4] Bourbaki, N. Lie Groups and Lie Algebras, Chapters 1–3, Springer, 1988

[5] Deligne, P. Théorie de Hodge, II, Publ. Math. Inst. Hautes Études Sci., Volume 40 (1971), pp. 5-57

[6] Enriquez, B. Flat connections on configuration spaces and braid groups of surfaces, Adv. Math., Volume 252 (2014), pp. 204-226

[7] Totaro, B. Configuration spaces of algebraic varieties, Topology, Volume 35 (1996) no. 4, pp. 1057-1067

Cited by Sources: