Torelli Theorem for the Deligne–Hitchin Moduli Space

2012

Let X and X' be compact Riemann surfaces of genus at least three. Let G and G' be nontrivial connected semisimple linear algebraic groups over C. If some components $M_{DH}^d(X,G)$ and $M_{DH}^{d'}(X',G')$ of the associated Deligne--Hitchin moduli spaces are biholomorphic, then X' is isomorphic to X or to the conjugate Riemann surface $\bar{X}$.

Communications in Contemporary Mathematics, 2009

Let X be any compact connected Riemann surface of genus g, with g ≥ 3. For any r ≥ 2, let [Formula: see text] denote the moduli space of holomorphic SL (r,ℂ)-connections over X. It is known that the biholomorphism class of the complex variety [Formula: see text] is independent of the complex structure of X. If g = 3, then we assume that r ≥ 3. We prove that the isomorphism class of the variety [Formula: see text] determines the Riemann surface X uniquely up to an isomorphism. A similar result is proved for the moduli space of holomorphic GL (r,ℂ)-connections on X. We also show that the Torelli theorem remains valid for the moduli spaces of connections, as well as those of stable vector bundles, on geometrically irreducible smooth projective curves defined over the field of real numbers.

Topology, 2007

Let (X , x 0) be any one-pointed compact connected Riemann surface of genus g, with g ≥ 3. Fix two mutually coprime integers r > 1 and d. Let M X denote the moduli space parametrizing all logarithmic SL(r, C)-connections, singular over x 0 , on vector bundles over X of degree d. We prove that the isomorphism class of the variety M X determines the Riemann surface X uniquely up to an isomorphism, although the biholomorphism class of M X is known to be independent of the complex structure of X. The isomorphism class of the variety M X is independent of the point x 0 ∈ X. A similar result is proved for the moduli space parametrizing logarithmic GL(r, C)-connections, singular over x 0 , on vector bundles over X of degree d. The assumption r > 1 is necessary for the moduli space of logarithmic GL(r, C)-connections to determine the isomorphism class of X uniquely.

Annales de l’institut Fourier, 2012

Let X and X ′ be compact Riemann surfaces of genus ≥ 3, and let G and G ′ be nonabelian reductive complex groups. If one component M d G (X) of the coarse moduli space for semistable principal G-bundles over X is isomorphic to another component M d ′ G ′ (X ′), then X is isomorphic to X ′ .

Let g be an integer ≥ 3 and let B g = {X ∈ M g | Aut X = e} , where M g denotes the moduli space of a compact Riemann surface. The geometric structure of B g is of substantial interest because B g corresponds to the singularities of the action of the modular group on the Teichmüller space of surfaces of genus g (see [H]). Surprisingly R.S. Kulkarni [K] has found isolated points in B g. He showed that they appear if and only if 2g + 1 is an odd prime distinct from 7. The aim of this paper is to find a geometrical explanation of this phenomenon using the fact that the isolated points are given by surfaces admitting anticonformal involutions (symmetries). The points in the Teichmüller space, corresponding to groups uniformizing surfaces with a symmetry, is a (non disjoint) union of submanifolds. We shall obtain that the isolated intersections of such submanifolds give us the isolated points in the branch loci. Also we prove that there are no isolated points in the moduli space of Klein surfaces which are not Riemann surfaces.

Annals of Mathematics, 2003

We give infinite series of groups Γ and of compact complex surfaces of general type S with fundamental group Γ such that 1) Any surface S ′ with the same Euler number as S, and fundamental group Γ, is diffeomorphic to S. 2) The moduli space of S consists of exactly two connected components, exchanged by complex conjugation. * The research of the author was performed in the realm of the SCHWERPUNKT "Globale Methode in der komplexen Geometrie", and of the EAGER EEC Project.

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2017

In 1962 E. H. Rauch found the points in the moduli space of Riemann surfaces not having a neighbourhood homeomorphic to a ball. These points are called here topologically singular. We give a different proof of some of the results of Rauch and also determine the topologically singular points in the branch locus of some equisymmetric families of Riemann surfaces.

Journal of Mathematical Sciences, 2009

We investigate the moduli space of complex structures on the Riemann sphere with marked points using signature formulas.

Manuscripta Mathematica, 1995

We study that subset of the moduli space Ma of stable genus g, g > 1, Riemann surfaces which consists of such stable Riemann surfaces on which a given finite group F acts. We show first that this subset is compact. It turns out that, for general finite groups F, the above subset is not connected. We show, however, that for Z2 actions this subset is connected. Finally, we show that even in the moduli space of smooth genus g Riemann surfaces, the subset of those Riemann surfaces on which Z2 acts is connected, ha view of deliberations of Klein ([8]), this was somewhat surprising. These results are based on new coordinates for moduli spaces. These coordinates are obtained by certain regular triangulations of Riemann surfaces. These triangulations play an important role also elsewhere, for instance in apl)roximating eigenfunctions of tim Laplace operator numerically.

Compositio Mathematica, 2004

Fix a τ-connection D L on a line bundle L over X × C, where X is a compact connected Riemann surface of genus at least three. Let M X (D L) denote the moduli space of all semistable τ-connections of rank n, where n 2, with the property that the induced τ-connection on the top exterior product is isomorphic to (L, D L). Let M Y (D M) be the similar moduli space for another Riemann surface Y with genus(Y) = genus(X), where D M is a τ-connection on a line bundle M over Y ×C. We prove that if the variety M X (D L) is isomorphic to M Y (D M), then X is isomorphic to Y. Let M D X denote the moduli space of all rank n flat connections on X. We prove that M D X determines X up to finitely many Riemann surfaces. For the very general Riemann surface X, the variety M D X determines X.