Parabolic Bundles Research Articles

On Faltings parabolic theta functions

In this article, we prove that the Faltings theta functions on moduli of parabolic bundles over a smooth projective curve can be used to give an explicit scheme-theoretic projective embedding.

Rank-level duality of conformal blocks for odd orthogonal Lie algebras in genus $0$

Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r+1) on the pointed projective line conjectured by T. Nakanishi and A. Tsuchiya.

Goldman Systems and Bending Systems

AbstractWe show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and as symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.

Flat Rank Two Vector Bundles on Genus Two Curves

We study the moduli space of trace-free irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of under- lying vector bundles (including unstable bundles), for which we compute a natural Lagrangian rational section. As a particularity of the genus 2 case, connections as above are invariant under the hyperelliptic involution : they descend as rank 2 logarithmic connections over the Riemann sphere. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, Tyurin and Bertram. This allow us to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical (16, 6)-configuration of the Kummer surface. We also recover a Poincare family due to Bolognesi on a degree 2 cover of the Narasimhan-Ramanan moduli space. We explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by vanGeemen-Previato. We explicitly describe the isomonodromic foliation in the moduli space of vector bundles with sl(2,C)-connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles.

Symplectic form on hyperpolygon spaces

In Godinho and Mandini (Adv Math 244:465–532, 2013), a family of parabolic Higgs bundles on \({{\mathbb {C}}}{\mathbb {P}}^1\) was constructed and the parameter space for the family was identified with a moduli space of hyperpolygons. Our aim here is to give a canonical alternative construction of this family. This enables us to compute the Higgs symplectic form for this family and show that the isomorphism of Godinho and Mandini (Adv Math 244:465–532, 2013) is actually a symplectomorphism.

CONNECTIONS ON REAL PARABOLIC BUNDLES OVER A REAL CURVE

We give analogous criterion to admit a real parabolic connection on real parabolic bundles over a real curve. As an application of this criterion, if real curve has a real point, then we proved that a real vector bundle E of rank r and degree d with gcd(r, d) = 1 is real indecomposable if and only if it admits a real logarithmic connection singular exactly over one point with residue given as multiplication by <TEX>$-\frac{d}{r}$</TEX>. We also give an equivalent condition for real indecomposable vector bundle in the case when real curve has no real points.

GRASSMANNIAN-FRAMED BUNDLES AND GENERALIZED PARABOLIC STRUCTURES

We build compact moduli spaces of Grassmannian-framed bundles over a Riemann surface, essentially replacing a group by a bi-equivariant compactification. We do this both in the algebraic and symplectic settings, and prove a Hitchin–Kobayashi correspondence between the two. The spaces are universal spaces for parabolic bundles (in the sense that all of the moduli can be obtained as quotients), and the reduction to parabolic bundles commutes with the correspondence. An analogous correspondence is outlined for the generalized parabolic bundles of Bhosle.

Lagrangian Fibrations in Duality on Moduli Spaces of Rank 2 Logarithmic Connections Over the Projective Line

We study the moduli space of logarithmic connections of rank 2 on the Riemann sphere minus n points with fixed spectral data. There are two natural Lagrangian maps: one towards apparent singularities of the associated fuchsian scalar equation, and another one towards moduli of parabolic bundles. We show that these are transversal and dual to each other. In case n=5, we recover the beautiful geometry of Del Pezzo surfaces of degree 4.

Polygons in Minkowski three space and parabolic Higgs bundles of rank 2 on $ \mathbb{C}{{\mathbb{P}}^1} $

Consider the moduli space of parabolic Higgs bundles (E, Φ) of rank two on ℂℙ1 such that the underlying holomorphic vector bundle for the parabolic vector bundle E is trivial. It is equipped with the natural involution defined by \( \left( {E,\varPhi } \right)\mapsto \left( {E,-\varPhi } \right) \). We study the fixed point locus of this involution. In [GM], this moduli space with involution was identified with the moduli space of hyperpolygons equipped with a certain natural involution. Here we identify the fixed point locus with the moduli spaces of polygons in Minkowski 3-space. This identification yields information on the connected components of the fixed point locus.

The Chern invariants for parabolic bundles at multiple points

Abstract If D ⊂ X is a curve with multiple points in a surface, a parabolic bundle defined on (X;D) away from the singularities can be extended in several ways to a parabolic bundle on a resolution of singularities. We investigate the possible parabolic Chern classes for these extensions.

Quantization of some moduli spaces of parabolic vector bundles on $${{\mathbb C}{\mathbb P}^1}$$

We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over $\mathbf{CP}^1$ with four parabolic points and parabolic weights in {0,1/2}. Identifying such parabolic bundles as vector bundles on an elliptic curve, we obtain explicit expressions for the corresponding non-abelian theta functions. These non-abelian theta functions are described in terms of certain naturally defined distributions on the compact group SU(2).

An example of the Langlands correspondence for irregular rank two connections on [formula omitted

Special kinds of rank 2 vector bundles with (possibly irregular) connections on P1 are considered. We construct an equivalence between the derived category of quasi-coherent sheaves on the moduli stack of such bundles and the derived category of modules over a TDO ring on certain non-separated curve. We identify this curve with the coarse moduli space of some parabolic bundles on P1. Then our equivalence becomes an example of the categorical Langlands correspondence.

Pullback of parabolic bundles and covers of ℙ1\{0,1,∞}

We work over an algebraically closed ground field of characteristic zero. A $G$-cover of ${\mathbb P}^1$ ramified at three points allows one to assign to each finite dimensional representation $V$ of $G$ a vector bundle $\oplus \mathscr{O}(s_i)$ on ${\mathbb P}^1$ with parabolic structure at the ramification points. This produces a tensor functor from representation of $G$ to vector bundles with parabolic structure that characterises the original cover. This work attempts to describe this tensor functor in terms of group theoretic data. More precisely, we construct a pullback functor on vector bundles with parabolic structure and describe the parabolic pullback of the previously described tensor functor.

PARABOLIC k-AMPLE BUNDLES

We construct projectivization of a parabolic vector bundle and a tautological line bundle over it. It is shown that a parabolic vector bundle is ample if and only if the tautological line bundle is ample. This allows us to generalize the notion of a k-ample bundle, introduced by Sommese, to the context of parabolic bundles. A parabolic vector bundle E* is defined to be k-ample if the tautological line bundle [Formula: see text] is k-ample. We establish some properties of parabolic k-ample bundles.

Orthogonal and symplectic parabolic bundles

We investigate orthogonal and symplectic bundles with parabolic structure, over a curve.

Torelli theorems for moduli of logarithmic connections and parabolic bundles

Let Z be a finite subset of a compact connected Riemann Surface X. Let \({\fancyscript{M}_X^{lc}}\) denote the moduli space of pairs (L, D) where L is a line bundle on X and D is a logarithmic connection on L singular along Z. Then \({\fancyscript{M}_X^{lc}}\) has a natural symplectic structure [ωX]. We show that the pair \({(\fancyscript{M}_X^{lc},[\omega_X])}\) determines X and there are no nonconstant algebraic functions on \({\fancyscript{M}_X^{lc}}\). We also prove a Torelli type theorem for the moduli space of parabolic bundles.

Determinant line bundle on moduli space of parabolic bundles

In Biswas and Raghavendra (Proc Indian Acad Sci (Math Sci) 103:41–71, 1993; Asian J Math 2:303–324, 1998), 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.

Rank-level duality for conformal blocks of the linear group

We generalise the rank-level duality (also known as strange duality) proved by P. Belkale [J. Amer. Math. Soc. 21 (2008), no. 1, 235–258] for generic curves and A. Marian and D. Oprea [Invent. Math. 168 (2007), no. 2, 225–247] for every smooth curve to the case of spaces of conformal blocks related to moduli spaces of parabolic bundles on a smooth pointed projective curve.

Differential forms on moduli spaces of parabolic bundles

Differential forms on moduli spaces of parabolic bundles

Kac's Theorem for weighted projective lines

We prove an analogue of Kac’s Theorem, describing the dimension types of indecomposable coherent sheaves (or parabolic bundles) over weighted projective lines in terms of root systems for loop algebras of Kac–Moody Lie algebras. We use a theorem of Peng and Xiao to associate a Lie algebra to the category of coherent sheaves for a weighted projective line over a finite field, and find elements of this Lie algebra which satisfy the relations defining the loop algebra. We use these elements in the proof of our analogue of Kac’s Theorem.