A Torelli theorem for the moduli space of parabolic Higgs bundles. (English) Zbl 1226.14018
The authors extend the proof of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant given by I. Biswas and T. L. Gómez [Q. J. Math. 54, No. 2, 159–169 (2003; Zbl 1043.14006)] to the parabolic situation.
The main result of the paper is Theorem 1.1. It states that a complex projective smooth curve \(X\) of genus \(g\geq 2\) with marked points can be recovered from the moduli space of stable parabolic Higgs bundles of rank \(2\) over \(X\) of small weights with fixed determinant of odd degree.
In order to prove this result, the authors apply the Torelli theorem from [Math. Proc. Camb. Philos. Soc. 130, No. 2, 269–280 (2001; Zbl 1063.14042)]. That is why one considers only the case of rank \(2\).
Indeed, the moduli space of parabolic Higgs bundles contains the moduli space of parabolic bundles (those with zero Higgs field). One can characterize this embedding intrinsically: the space of parabolic bundles corresponds to the only irreducible component of the fibre of the Hitchin map over the origin of the Hitchin space that does not admit a non-trivial \(\mathbb C^*\)-action.
Using the ideas from [Q. J. Math. 54, No. 2, 159–169 (2003; Zbl 1043.14006)], given the moduli space of parabolic Higgs bundles as an abstract algebraic variety, the authors recover the Hitchin map. Using the Kodaira-Spencer map it is possible to recover the origin of the Hitchin space. So, once there is an isomorphism of two moduli spaces of parabolic Higgs bundles, it yields an isomorphism of the moduli spaces of parabolic bundles, which are subvarieties as explained above. Hence one can apply the result from [Math. Proc. Camb. Philos. Soc. 130, No. 2, 269–280 (2001; Zbl 1063.14042)].
The paper under review consists of 6 sections.
Section 1 is an introduction. A short overview of the history of the subject and some relevant references are presented here. The main result of the paper, Theorem 1.1, is stated here as well together with an outline of the paper. In Section 2 some necessary preliminaries on parabolic Higgs bundles are given. Section 3 provides the definitions and basic properties of the Hitchin space and the Hitchin map for parabolic Higgs bundles. In Section 4 the properties of the nilpotent cone, i. e., the fibre of the Hitchin map over \(0\), are discussed. The Kodaira-Spencer map is used for characterizing the origin of the Hitchin space in Section 5. Section 6 provides a proof of the main result of the paper.
The main result of the paper is Theorem 1.1. It states that a complex projective smooth curve \(X\) of genus \(g\geq 2\) with marked points can be recovered from the moduli space of stable parabolic Higgs bundles of rank \(2\) over \(X\) of small weights with fixed determinant of odd degree.
In order to prove this result, the authors apply the Torelli theorem from [Math. Proc. Camb. Philos. Soc. 130, No. 2, 269–280 (2001; Zbl 1063.14042)]. That is why one considers only the case of rank \(2\).
Indeed, the moduli space of parabolic Higgs bundles contains the moduli space of parabolic bundles (those with zero Higgs field). One can characterize this embedding intrinsically: the space of parabolic bundles corresponds to the only irreducible component of the fibre of the Hitchin map over the origin of the Hitchin space that does not admit a non-trivial \(\mathbb C^*\)-action.
Using the ideas from [Q. J. Math. 54, No. 2, 159–169 (2003; Zbl 1043.14006)], given the moduli space of parabolic Higgs bundles as an abstract algebraic variety, the authors recover the Hitchin map. Using the Kodaira-Spencer map it is possible to recover the origin of the Hitchin space. So, once there is an isomorphism of two moduli spaces of parabolic Higgs bundles, it yields an isomorphism of the moduli spaces of parabolic bundles, which are subvarieties as explained above. Hence one can apply the result from [Math. Proc. Camb. Philos. Soc. 130, No. 2, 269–280 (2001; Zbl 1063.14042)].
The paper under review consists of 6 sections.
Section 1 is an introduction. A short overview of the history of the subject and some relevant references are presented here. The main result of the paper, Theorem 1.1, is stated here as well together with an outline of the paper. In Section 2 some necessary preliminaries on parabolic Higgs bundles are given. Section 3 provides the definitions and basic properties of the Hitchin space and the Hitchin map for parabolic Higgs bundles. In Section 4 the properties of the nilpotent cone, i. e., the fibre of the Hitchin map over \(0\), are discussed. The Kodaira-Spencer map is used for characterizing the origin of the Hitchin space in Section 5. Section 6 provides a proof of the main result of the paper.
Reviewer: Oleksandr Iena (Luxembourg)
MSC:
| 14D22 | Fine and coarse moduli spaces |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14H60 | Vector bundles on curves and their moduli |
References:
| [1] | Mumford, Math Newstead Periods of a moduli space of bundles on curves Ramanan Deformations of the moduli space of vector bundles over an algebraic curve Ramanan Generalised Prym varieties as fixed points Indian Math, Ann Math Math 38 pp 248– (1980) |
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