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

Determinant line bundle on moduli space of parabolic bundles

Annals of Global Analysis and Geometry, 2011

In [BR1], [BR2], a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line bundle on a moduli space of usual vector bundles over a covering curve. The Hermitian structure on the parabolic determinant bundle was taken to be the pullback of the Quillen metric on the determinant line bundle on the moduli space of usual vector bundles. Here a direct construction of the Hermitian structure is given. For that we need to establish a version of the correspondence between the stable parabolic bundles and the Hermitian-Einstein connections in the context of conical metrics. Also, a recently obtained parabolic analog of Faltings' criterion of semistability plays a crucial role.

Determinants of parabolic bundles on Riemann surfaces

Proceedings Mathematical Sciences

Let X be a compact Riemann surface and M~(X) the moduli space of stable parabolic vector bundles with fixed rank, degree, rational weights and multiplicities. There is a natural K/ihler metric on MP,(X). We obtain a natural metrized holomorphic line bundle on M~(X) whose Chern form equals mr times the K/ihler form, where m is the common denominator of the weights and r the rank.

Some remarks on the local moduli of tangent bundles over complex surfaces

American Journal of Mathematics, 2003

Hirzebruch on the occasion of his seventy-fifth birthday. His celebrated work on the Riemann-Roch formula and mathematical insights have greatly influenced our work. Abstract. Using the Hirzebruch's Riemann-Roch formula for endomorphism bundles over a compact complex twofold we prove that the tangent bundle of a complex surface M of general type admits a nontrivial trace-free deformation, unless M is holomorphically covered by the euclidean ball. It follows that the tangent bundle of the Mostow-Siu surface, which is a Kahler surface with a negative definite curvature tensor, does have a nontrivial trace-free moduli. Among some other results we also point out a relationship between the Kuranishi obstruction and symmetric holomorphic two tensors on a complex surface.

Vector bundles and connections on Riemann surfaces with projective structure

2021

Let Bg(r) be the moduli space of triples of the form (X, K X , F ), where X is a compact connected Riemann surface of genus g, with g ≥ 2, K X is a theta characteristic on X , and F is a stable vector bundle on X of rank r and degree zero. We construct a T Bg(r)–torsor Hg(r) over Bg(r). This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank r, on a fixed Riemann surface Y , given by the moduli space of holomorphic connections on the stable vector bundles of rank r on Y , and on the other hand the torsor over the moduli space of Riemann surfaces given by the moduli space of Riemann surfaces with a projective structure. It is shown that Hg(r) has a holomorphic symplectic structure compatible with the T Bg(r)–torsor structure. We also describe Hg(r) in terms of the second order matrix valued differential operators. It is shown that Hg(r) is identified with the T Bg(r)–torsor given by the sheaf of holomorphic connections on the theta line bund...

A Torelli theorem for the moduli space of parabolic rank two vector bundles over curves

Let S (respectively, S') be a finite subset of a compact connected Riemann surface X (respectively, X') of genus at least two. Let M (respectively, M') denote a moduli space of parabolic stable bundles of rank two over X (respectively, X') with fixed determinant of degree one,having a nontrivial quasi-parabolic structure over each point of S (respectively, S'), and of parabolic degree less than two. It is proved that M is isomorphic to M' if and only if there is an isomorphism of X with X' taking S to S'.

Chern forms and the Riemann tensor for the moduli space of curves

Inventiones Mathematicae, 1986

Two vector bundles associated to the moduli space of compact Riemann surfaces have a Hermitian metric derived from the hyperbolic geometry of Riemann surfaces. Briefly our purpose is to determine the connection and curvature forms for these metrics.

Some index formulae on the moduli space of stable parabolic vector bundles

2008

We study natural families of ∂-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.

Stable vector bundles on the families of curves

European Journal of Mathematics, 2022

We offer a new approach to proving the Chen–Donaldson–Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a family of curves and its relation to the one-dimensional cycles in the moduli space of stable bundles on curves.