Monodromy of Schwarzian equations with regular singularities

Faraco, Gianluca¹ ; Gupta, Subhojoy²

¹Dipartimento di Matematica e Applicazioni U5, Università degli Studi di Milano-Bicocca, Milan, Italy
²Department of Mathematics, Indian Institute of Science, Bangalore, India

Geometry & topology, Tome 29 (2025) no. 2, pp. 549-617

Résumé

Let S be a punctured surface of finite type and negative Euler characteristic. We determine all possible representations ρ: π1(S) → PSL ⁡ 2(ℂ) that arise as the monodromy of the Schwarzian equation on S with regular singularities at the punctures. Equivalently, we determine the holonomy representations of complex projective structures on S whose Schwarzian derivatives, with respect to some uniformizing structure, have poles of order at most two at the punctures. Following earlier work that dealt with the case when there are no apparent singularities, our proof reduces to the case of realizing a degenerate representation with apparent singularities. This mainly involves explicit constructions of complex affine structures on punctured surfaces, with prescribed holonomy. As a corollary, we determine the representations that arise as the holonomy of spherical metrics on S with cone points at the punctures.

Détail

DOI : 10.2140/gt.2025.29.549

Keywords: Schwarzian equations, complex projective structures

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