Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space

Biswas, Indranil

doi:10.5802/aif.1584

Cited by 4 publications

(5 citation statements)

References 3 publications

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“…To see this action, note that using the earlier mentioned identification N X = R g (defined in Equation (2.1)), the fiber bundle P in Equation (2.3) has a natural flat connection (this flat connection is not holomorphic). The monodromy of this flat connection gives an action of Γ 1 g = π 1 (M 1 g , x 0 ) on N X ; more details can be found in [2]. The action of Γ 1 g on H i (N X , Z) induced by the above action of Γ 1 g on N X evidently coincides with the monodromy representation of the local system…”

Section: This Group γmentioning

confidence: 91%

“…(2.3) has a natural flat connection (this flat connection is not holomorphic). The monodromy of this flat connection gives an action of Γ 1 g = π 1 (M 1 g , x 0 ) on N X ; more details can be found in [2]. The action of Γ 1 g on H i (N X , Z) induced by the above action of Γ 1 g on N X evidently coincides with the monodromy representation of the local system R i P * Z on M 1 g , where Z is the constant local system on N with stalk Z.…”

Section: Introductionmentioning

confidence: 99%

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On the rational homotopy type of a moduli space of vector bundles over a curve

Biswas¹,

Muñoz²

2008

Communications in Analysis and Geometry

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We study the rational homotopy of the moduli space N X that parametrizes the isomorphism classes of all stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g, with g ≥ 2. The symplectic group Aut(H 1 (X, Z)) ∼ = Sp(2g, Z) has a natural action on the rational homotopy groups π n (N X )⊗ Z Q. We prove that this action extends to an action of Sp(2g, C) on π n (N X )⊗ Z C. We also show that π n (N X )⊗ Z C is a non-trivial representation of Sp(2g, C) ∼ = Aut (H 1 (X, C)) for all n ≥ 2g − 1. In particular, N X is a rationally hyperbolic space. In the special case where g = 2, for each n ∈ N, we compute the leading Sp(2g, C) representation occurring in π n (N X ) ⊗ Z C.

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Section: This Group γmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

On the rational homotopy type of a moduli space of vector bundles over a curve

Biswas¹,

Muñoz²

2008

Communications in Analysis and Geometry

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“…Consider the relative (1, l)-form 0T considered in Theorem 3.12. Using the projection 7 in (3.9), the relative form QT gives a (1, l)-form on N^, which we shall also denote by QT-From Theorem 5.4 of [6] we know that the curvature of the hermitian connection on LQ is of the form:…”

Section: Sr = 7r 1 (X-d)/7r 1 {Z-h-1 (D))mentioning

confidence: 99%

“…For each parameter t G T, the fiber F~l(t) is the moduli space of stable vector bundles of rank r and degree d over Xt = /~1(t). There is a universal adjoint bundle over Mr-In [6] one of the authors works with the determinant of this universal adjoint bundle, constructs a natural Hermitian metric on this determinant line bundle and computes its curvature.…”

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confidence: 99%

Curvature of the determinant bundle and the Kähler form over the moduli of parabolic bundles for a family of pointed curves

Biswas

Raghavendra²

1998

Asian Journal of Mathematics

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“…It carries a universal theta divisor, also denoted by Θ X , which specializes to the theta divisor on each M X (n). (See [9] for a discussion of this moduli space. )…”

Section: Extended Higgs Fieldsmentioning

confidence: 99%

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Ben‐Zvi¹,

Biswas²

2003

Internat. Math. Res. Notices

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The nonabelian prime formWe focus on two different mechanisms linking our protagonists. The first and most direct link is given by difference maps. To a bundle E, is naturally associated the family E(y − x) of vector bundles on X, parametrized by (x, y) ∈ X × X. This family is classified by a map δ E from X × X to the moduli space of bundles. The pullback of the theta function by δ E is a canonical section of the dual determinant line bundle of this family. In Theorem 4.3, we describe this line bundle canonically in terms of E. The identification, the "nonabelian prime form," holds for arbitrary families of curves and bundles on them (i.e., over the moduli stack of curves and bundles, Corollary 4.5).In the case of line bundles, this identification is given by the classical Klein prime form. Moreover, up to a scalar and multiplication by this prime form, the abelian Szegö kernel and the pullback of the theta function are in fact equal. (This is sometimes used as the definition of the Szegö kernel.) In Theorem 4.6, we provide a cohomological description of the nonabelian Szegö kernel and of the nonabelian prime form (for E with no sections). This makes it easy to compare to the (cohomologically defined) theta function the prime form. In the higher rank case the pullback of the theta function is identified with the determinant of the Szegö kernel. (See also Remark 1.1 for related work.) Twisted cotangent bundlesThe second mechanism to relate the theta function and Szegö kernel is through a geometric description of the differential of the theta function, in terms of connection operators on the curve. More precisely, we identify the twisted forms of the cotangent bundle which carry the logarithmic differential of the theta function with spaces of kernel functions, as well as the corresponding sections. This is done by describing the behavior of the Szegö kernel along the generalized theta divisor. Let L denote a line bundle on a complex manifold M, and Conn M (L) the sheaf of holomorphic connections on L. Since the difference between any two (locally defined) connections on L is a holomorphic one-form, Conn M (L) forms an affine bundle (torsor) for the cotangent bundle Ω 1 M . Such an affine bundle, equipped with a compatible symplectic form (which for Conn M (L) arises as the curvature of a tautological connection), is known as a twisted cotangent bundle for M (see [2]). A meromorphic section s of L provides a meromorphic section of Conn M (L), which is the coordinate-free form of the logarithmic differential d log s. Twisted cotangent bundles on moduli spaces of curves and of bundles have been studied in [3, 5, 12, 26]. It is known that the twisted cotangent at Harvard Library on July 13, 2015 http://imrn.oxfordjournals.org/ Downloaded from Theta Functions and Szegö Kernels 1307 at Harvard Library on July 13, 2015 http://imrn.oxfordjournals.org/ Downloaded from −1/2 X (so the operation E → E 0 at Harvard Library on July 13, 2015 http://imrn.oxfordjournals.org/ Downloaded from −1/2 X , (2.5) since tensoring by a line...

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Determinant bundle over the universal moduli space of vector bundles over the Teichmüller space

Cited by 4 publications

References 3 publications

On the rational homotopy type of a moduli space of vector bundles over a curve

On the rational homotopy type of a moduli space of vector bundles over a curve

Curvature of the determinant bundle and the Kähler form over the moduli of parabolic bundles for a family of pointed curves

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