A Torelli type theorem for the moduli space of rank two connections on a curve

We look at rank two parabolic Higgs bundles over the projective line minus five points which are semistable with respect to a weight vector µ ∈ [0, 1] 5 . The moduli space corresponding to the central weight µ c = ( 1 2 , . . . , 1 2 ) is studied in details and all singular fibers of the Hitchin map are described, including the nilpotent cone. After giving a description of fixed points of the C * -action we obtain a proof of Simpson's foliation conjecture in this case. For each n ≥ 5, we remark that there is a weight vector so that the foliation conjecture in the moduli space of rank two logarithmic connections over the projective line minus n points is false.

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On the moduli of logarithmic connections on elliptic curves

Fassarella

Loray

Muniz

2022

Math. Z.

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

Biswas

Muñoz

2007

Topology

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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.

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A Torelli type theorem for the moduli space of rank two connections on a curve

Cited by 2 publications

References 13 publications

On the moduli of logarithmic connections on elliptic curves

On the moduli of logarithmic connections on elliptic curves

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

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