On Kawai theorem for orbifold Riemann surfaces

Takhtajan, Leon A.

doi:10.1007/s00208-018-1740-6

Cited by 3 publications

(9 citation statements)

References 24 publications

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“…The monodromy of a projective connection determines a natural map of P g to the PSL(2, C)-character variety, allowing to pull back the Goldman symplectic form to P g . As stated in [10], these two pullbacks give the same (up to a constant) symplectic form on P g (see [21] for a direct proof). According to a theorem of Goldman [7], T g is naturally isomorphic to the component of the PSL(2, R)-character variety with the maximal Euler class, and the pullback of the Goldman form to the Teichmüller space is (up to a constant) the Weil-Petersson symplectic form on T g [6].…”

Section: Introductionmentioning

confidence: 76%

“…The fundamental class [X] can be realized as the following 2-cycle in the group homology (see [6,21] and references therein)…”

Section: Eichler Integralsmentioning

confidence: 99%

“…Remark 5. The proof of Theorem 3 is adapted for the case of vector bundles simplified version of the proof of Theorem 1 in [21].…”

Section: Pullback Of Goldman Formmentioning

confidence: 99%

“…Theorem 3 is analogue of 'Kawai theorem' -a theorem in [10] that the pullback by the monodromy map of the Goldman symplectic form on the PSL(2, C)-character variety to the bundle of projective structures on the Riemann surfaces, identified with the holomorphic cotangent bundle to the Teichmüller space by the Bers section, coincides with the Liouville symplectic form. However, the proof of formula (5.4) in [10] is not correct since the functions ẇν (z) on H (using Ahlfors notation as in [10,21]) do not transform like vector fields under the group Γ. For harmonic ν formula (5.4) immediately follows from Ahlfors formula that ẇν zzz (z) = 0 and (5.4) trivially holds, while for general ν it is precisely the quasi-Fuchsian reciprocity, proved in [13,19].…”

Section: Pullback Of Goldman Formmentioning

confidence: 99%

“…In § 2.4 we review Eichler integrals for harmonic (0, 1)-forms, used in Lemma 2 in § 2.5 for the explicit solution of the infinitesimal form of the equation (2.3). In § 3.1 we briefly discuss the Goldman symplectic form on the character variety, referring to [6,21] for the details. For convenience of the reader, in § 3.2 we prove Theorem 1, the Riemann bilinear relations for the matrix-valued 1-forms expressed in terms of the Eichler-Shimura periods, and as a corollary obtain Theorem 2, that the pullback to N of the Goldman form on the U(n)-character variety is the Narasimhan-Atiyah-Bott symplectic form.…”

Section: Introductionmentioning

confidence: 99%

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Goldman form, flat connections and stable vector bundles

Takhtajan¹

2021

Preprint

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We consider the moduli space N of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle A → N of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of of the quasi-Fuchsian projective connections -local holomorphic sections of A -that allow to pull back the Liouville symplectic form on T * N to A . We prove that the pullback of the Goldman form to A by the Riemann-Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result -the pullback of Goldman symplectic form to N by the Narasimhan-Seshadri connection is the natural symplectic form on N , introduced by Narasimhan and Atiyah & Bott.

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Section: Introductionmentioning

confidence: 76%

“…The fundamental class [X] can be realized as the following 2-cycle in the group homology (see [6,21] and references therein)…”

Section: Eichler Integralsmentioning

confidence: 99%

“…Remark 5. The proof of Theorem 3 is adapted for the case of vector bundles simplified version of the proof of Theorem 1 in [21].…”

Section: Pullback Of Goldman Formmentioning

confidence: 99%

Section: Pullback Of Goldman Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Goldman form, flat connections and stable vector bundles

Takhtajan¹

2021

Preprint

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Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Bertola,

Norton,

Ruzza

2024

Nonlinearity

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We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface of genus g ⩾ 2 with the explicit parametrization in terms of the Tyurin data. The ‘non-abelian’ theta divisor consists of bundles such that h 1 ⩾ 1 . On the complement of this divisor we construct a non-abelian (i.e. matrix) Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which also depends holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the G L n character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h 1.

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Goldman form, flat connections and stable vector bundles

Takhtajan

2022

Enseign. Math.

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We consider the moduli space \mathscr{N} of stable vector bundles of degree 0 over a compact Riemann surface and the affine bundle \mathscr{A}\to \mathscr{N} of flat connections. Following the similarity between the Teichmüller spaces and the moduli of bundles, we introduce the analogue of the quasi-Fuchsian projective connections – local holomorphic sections of \mathscr{A} – that allow to pull back the Liouville symplectic form on T*\mathscr{N} to \mathscr{A} . We prove that the pullback of the Goldman form to \mathscr{A} by the Riemann–Hilbert correspondence coincides with the pullback of the Liouville form. We also include a simple proof, in the spirit of Riemann bilinear relations, of the classic result – the pullback of Goldman symplectic form to \mathscr{N} by the Narasimhan–Seshadri connection is the natural symplectic form on \mathscr{N} , introduced by Narasimhan and Atiyah & Bott.

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On Kawai theorem for orbifold Riemann surfaces

Cited by 3 publications

References 24 publications

Goldman form, flat connections and stable vector bundles

Goldman form, flat connections and stable vector bundles

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

Goldman form, flat connections and stable vector bundles

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