TY - JOUR

T1 - Quillen connection and the uniformization of Riemann surfaces

AU - Biswas, Indranil

AU - Favale, Filippo Francesco

AU - Pirola, Gian Pietro

AU - Torelli, Sara

N1 - Funding Information: We would like to thank Alessandro Ghigi for helpful remarks and discussions on the topic of the article. The second and third named authors are partially supported by INdAM - GNSAGA. The second named author is partially supported by “2017-ATE-0253” (Dipartimento di Matematica e Applicazioni - Università degli Studi di Milano-Bicocca). The third named author is partially supported by PRIN 2017“Moduli spaces and Lie theory” and by (MIUR): Dipartimenti di Eccellenza Program (2018-2022) - Dept. of Math. Univ. of Pavia. The first named author is partially supported by a J. C. Bose Fellowship.

PY - 2022/12

Y1 - 2022/12

N2 - The Quillen connection on \({\mathcal L} \rightarrow {\mathcal M}_g\), where \({\mathcal L}^*\) is the Hodge line bundle over the moduli stack of smooth complex projective curves curves \({\mathcal M}_g\), \(g \geq 5\), is uniquely determined by the condition that its curvature is the Weil--Petersson form on \({\mathcal M}_g\). The bundle of holomorphic connections on \({\mathcal L}\) has a unique holomorphic isomorphism with the bundle on \({\mathcal M}_g\) given by the moduli stack of projective structures. This isomorphism takes the \(C^\infty\) section of the first bundle given by the Quillen connection on \({\mathcal L}\) to the \(C^\infty\) section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.

AB - The Quillen connection on \({\mathcal L} \rightarrow {\mathcal M}_g\), where \({\mathcal L}^*\) is the Hodge line bundle over the moduli stack of smooth complex projective curves curves \({\mathcal M}_g\), \(g \geq 5\), is uniquely determined by the condition that its curvature is the Weil--Petersson form on \({\mathcal M}_g\). The bundle of holomorphic connections on \({\mathcal L}\) has a unique holomorphic isomorphism with the bundle on \({\mathcal M}_g\) given by the moduli stack of projective structures. This isomorphism takes the \(C^\infty\) section of the first bundle given by the Quillen connection on \({\mathcal L}\) to the \(C^\infty\) section of the second bundle given by the uniformization theorem. Therefore, any one of these two sections determines the other uniquely.

KW - Projective structure

KW - Quillen connection

KW - Torsor

KW - Uniformization

UR - http://www.scopus.com/inward/record.url?scp=85130215883&partnerID=8YFLogxK

U2 - 10.1007/s10231-022-01220-y

DO - 10.1007/s10231-022-01220-y

M3 - Article

VL - 201

SP - 2825

EP - 2835

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

SN - 0373-3114

IS - 6

ER -