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Abstract

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...

The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations

“…We can define a natural map from the moduli space of extended connections to the moduli space of pairs (C, E). This map is generalization of the map (E, ∇) → E and has been investigated in [5,Section 6] and [7,Section 4.3]. In this section, we consider parabolic connections instead of holomorphic connections and study the moduli space of parabolic connections with a quadratic differential instead of the moduli space of extended connections.…”

“…Put D(t) = t 1 + · · · + t n . We describe a description of (t, ν)parabolic connection with a quadratic differential in terms of a "integral kernel" on C ×C as in [5] and [6]. Let p 1 : C ×C → C and p 2 : C ×C → C be the first and second projections, respectively.…”

“…The map from the moduli space of pairs (E, ∇) to the moduli space of vector bundles defined by (E, ∇) → E is a twisted cotangent bundle on the moduli space of vector bundles. Here, E is a rank r vector bundle on the fixed curve C and ∇ is a holomorphic connection on E. This twisted cotangent bundle has been investigated by Faltings, Ben-Zvi-Biswas, and Ben-Zvi-Frenkel (see [10,Section 4], [8, Section 5], [5,Section 5], and [7, Section 4.1]). Moreover, Ben-Zvi-Biswas and Ben-Zvi-Frenkel studied on a twisted cotangent bundle on the moduli space of pairs (C, E) (see [5,Section 6] and [7,Section 4.3]).…”

“…Here, E is a rank r vector bundle on the fixed curve C and ∇ is a holomorphic connection on E. This twisted cotangent bundle has been investigated by Faltings, Ben-Zvi-Biswas, and Ben-Zvi-Frenkel (see [10,Section 4], [8, Section 5], [5,Section 5], and [7, Section 4.1]). Moreover, Ben-Zvi-Biswas and Ben-Zvi-Frenkel studied on a twisted cotangent bundle on the moduli space of pairs (C, E) (see [5,Section 6] and [7,Section 4.3]). In [5,6], Ben-Zvi and Biswas have introduced extended connections, which are generalization of holomorphic connections.…”

“…Moreover, Ben-Zvi-Biswas and Ben-Zvi-Frenkel studied on a twisted cotangent bundle on the moduli space of pairs (C, E) (see [5,Section 6] and [7,Section 4.3]). In [5,6], Ben-Zvi and Biswas have introduced extended connections, which are generalization of holomorphic connections. We can define a natural map from the moduli space of extended connections to the moduli space of pairs (C, E).…”