Line bundles over a moduli space of logarithmic connections on a Riemann surface

We consider logarithmic connections, on rank n and degree d vector bundles over a compact Riemann surface X, singular over a fixed point x₀ ∈ X with residue in the center of the integers n and d are assumed to be mutually coprime. A necessary and sufficient condition is given for a vector bundle to admit such a logarithmic connection. We also compute the Picard group of the moduli space of all such logarithmic connections. Let denote the moduli space of all such logarithmic connections, with the underlying vector bundle being of fixed determinant L, and inducing a fixed logarithmic connection on the determinant line L. Let be the Zariski open dense subset parametrizing all connections such that the underlying vector bundle is stable. The space of all global sections of certain line bundles on are computed. In particular, there are no nonconstant algebraic functions on Therefore, there are no nonconstant algebraic functions on although is biholomorphic to a representation space which admits nonconstant algebraic functions. The moduli space admits a natural compactification by a smooth divisor. We investigate numerically effectiveness of this divisor at infinity. It turns out that the divisor is not numerically effective in general. Received: March 2004 Revision: May 2004 Accepted: May 2004     

Cohomology of the moduli of parabolic vector bundles

The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact). While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan strata. If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free in its cohomology. The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.     

The Torelli theorem for the moduli spaces of connections on a Riemann surface

Let (X,x₀) be any one-pointed compact connected Riemann surface of genus g, with g≥3. Fix two mutually coprime integers r>1 and d. Let M_X denote the moduli space parametrizing all logarithmic -connections, singular over x₀, on vector bundles over X of degree d. We prove that the isomorphism class of the variety M_X determines the Riemann surface X uniquely up to an isomorphism, although the biholomorphism class of M_X is known to be independent of the complex structure of X. The isomorphism class of the variety M_X is independent of the point x₀∈X. A similar result is proved for the moduli space parametrizing logarithmic -connections, singular over x₀, on vector bundles over X of degree d. The assumption r>1 is necessary for the moduli space of logarithmic -connections to determine the isomorphism class of X uniquely.     

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen’s metric and interprete it as a local index theorem for the family of \(\overline\partial\)-operators in the associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kähler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein–Maass series. The cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten’s volume computation.     

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.     

Chen-Ruan cohomology of some moduli spaces of parabolic vector bundles

Let (X,D) be an ?-pointed compact Riemann surface of genus at least two. For each point x∈D, fix parabolic weights such that . Fix a holomorphic line bundle ξ over X of degree one. Let PMξ denote the moduli space of stable parabolic vector bundles, of rank two and determinant ξ, with parabolic structure over D and parabolic weights . The group of order two line bundles over X acts on PMξ by the rule E_∗⊗L?E_∗⊗L. We compute the Chen-Ruan cohomology ring of the corresponding orbifold.     

Betti Numbers of parabolic U(2,1)-Higgs bundles moduli spaces

Let X be a compact Riemann surface together with a finite set of marked points. We use Morse theoretic techniques to compute the Betti numbers of the parabolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked point showing that the Poincaré polynomials depend on the system of weights of the parabolic bundle.      

Restriction theorems for homogeneous bundles

We prove that for an irreducible representation , the associated homogeneous -vector bundle Wτ is strongly semistable when restricted to any smooth quadric or to any smooth cubic in , where k is an algebraically closed field of characteristic ≠2,3 respectively. In particular Wτ is semistable when restricted to general hypersurfaces of degree?2 and is strongly semistable when restricted to the generic hypersurface of degree?2.     

Restriction theorems for principal bundles

Let E be a semistable (or stable) principal bundle over a smooth complex projective variety X, and let DX be a complete intersection. We study the (semi)stability of the restriction E| _D . Some of the results known for vector bundles, such as Grauert–Mülich, Flenner and Mehta–Ramanathan theorems, are generalized to principal bundles. Mathematics Subject Classification (2000): 14F05, 32L05The authors are members of VBAC (Vector Bundles on Algebraic Curves), which is partially supported by EAGER (EC FP5 Contract no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101). T.G. was supported by a postdoctoral fellowship of Ministerio de Educación y Cultura (Spain), and wants to thank the Tata Institute of Fundamental Research, where this work was done while he was a postdoctoral student.     

Monads and moduli of vector bundles

Horrocks has shown that every vector bundle on ₂ and ₃ admits a certain double-ended resolution by line bundles, which he called a monad. We reprove Horrocks' results taking much care of uniqueness of the monads so obtained. This technique should be useful for constructing moduli spaces of stable vector bundles.     

Local moduli of holomorphic bundles

We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.     

Equivariant bundles and connections

Let X be a connected complex manifold equipped with a holomorphic action of a complex Lie group G. We investigate conditions under which a principal bundle on X admits a G-equivariance structure.