The Torelli theorem for the moduli spaces of connections on a Riemann surface

Abstract

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.

Moduli space of rank one logarithmic connections over a compact Riemann surface

“…The moduli space of logarithmic connections over a complex projective variety singular over a smooth normal crossing divisor has been constructed in [33]. Several algebro-geometric invariants like Picard group, algebraic functions of the moduli space of holomorphic and logarithmic connections have been studied, see [7], [9], [10], [36] [27], and [26].…”

“…Let Θ be the canonical polarisation on the second intermediate Jacobian lc (X) of logarithmic connections singular exactly over one point x 0 of the compact Riemann surface X with fixed determinant such that the principally polarised abelian variety (J 2 (M x 0 lc (X)), Θ) is isomorphic to the principally polarised abelian variety (J 2 (U(n, L)), Θ). Imitating the similar technique as in [10,Section 4], a principal polarisation can be constructed on M lc (X, L). Proposition 5.9.…”

“…Then we have a natural generalisation of [9,p.797,Theorem 4.3], and the same ideas can be used to prove the following. In [7, Theorem 5.2], Torelli type theorem has been proved for the moduli space of holomorphic connections over compact Riemann surface, and in [10], Torelli type theorems have been proved for the moduli space of logarithmic connections singular exactly over one point with fixed residue. In this section, we prove the Torelli type theorem for the moduli spaces M lc (n, d) and M lc (n, L).…”

Brauer Group of Moduli of Higgs Bundles and Connections

“…Therefore, the morphism p in (2.10) makes U a torsor over N for the vector bundle Ω 1 N . From [BM1,p. (2.11) this follows from "Cohomological purity" [Mi,p. 241,Theorem VI.5.1] (it also follows from [Gr2,).…”

Torelli theorem for the moduli spaces of pairs

“…Proof. For M = M s X (n, L 0 ) the polarisation is constructed as follows (see [1, Section 8]; a similar argument is in [3,Section 4]). Let M = M X (n, L 0 ).…”