Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.

     

A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

Let $X$ be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation $\rho\colon \mathop{\rm GL}(r)\rightarrow \mathop{\rm GL}(V)$. Then one can associate to every vector bundle $E$ of rank $r$ over $X$ a vector bundle $E_\rho$ with fibre $V$. We would like to study triples $(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ over $X$, $L$ is a line bundle over $X$, and $\phi\colon E_\rho\rightarrow L$ is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert--Mumford criterion, and we establish the existence of moduli spaces for the semistable objects. In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.     

The Boden-Hu conjecture holds precisely up to rank 8

We consider moduli schemes of vector bundles over a smooth projective curve endowed with parabolic structures over a marked point. Boden and Hu observed that a slight variation of the weights leads to a desingularisation of the moduli scheme, and they conjectured that one can always obtain a small resolution this way. The present text proves this conjecture in some cases (including all bundles of rank up to eight) and gives counterexamples in all other cases (in particular in every rank beyond eight). The main tool is a generalisation of Ext-groups involving more than two quasiparabolic bundles.Mathematics Subject Classification (2000): 14H60, 14D20     

Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface M endowed with an anti-holomorphic involution which determines topologically the original surface S. In this paper, we compare dianalytic vector bundles over S and holomorphic vector bundles over M, devoting special attention to the implications that this has for moduli varieties of semistable vector bundles over M. We construct, starting from S, totally real, totally geodesic, Lagrangian submanifolds of moduli varieties of semistable vector bundles of fixed rank and degree over M. This relates the present work to the constructions of Ho and Liu over non-orientable compact surfaces with empty boundary (Ho and Liu in Commun Anal Geom 16(3):617–679, 2008).     

Vanishing of Chern Classes of the De Rham Bundles for Some Families of Moduli Spaces

Given a family of nonsingular complex projective surfaces, there is a corresponding family of Hilbert schemes of zero dimensional subschemes. We prove that the Chern classes, with values in the rational Chow groups, of the de Rham bundles for such a family of Hilbert schemes vanish. A similar result is proved for any relative moduli space of rank one sheaves with trivial integral first Chern class over any family of complex projective surfaces.     

Moduli for principal bundles over algebraic curves: II

We classify principal bundles on a compact Riemann surface. A moduli space for semistable principal bundles with a reductive structure group is constructed using Mumford’s geometric invariant theory. This is the second and concluding part of the thesis of late Professor A Ramanathan; the first part was published in the previous issue.     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

On some moduli spaces of stable vector bundles on cubic and quartic threefolds

We study certain moduli spaces of stable vector bundles of rank 2 on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.     

On Framed Instanton Bundles and Their Deformations

We consider a compact twistor space P and assume that there is a surface S ⊂ P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space ([20], [18]). Similar to [6] and [5] we examine the restriction of an instanton bundle V equipped with a fixed trivialization along F to a framed vector bundle over (S, F). First we develope inspired by [13] a suitable deformation theory for vector bundles over an analytic space framed by a vector bundle over a subspace of arbitrary codimension. In the second section we describe the restriction as a smooth natural transformation into a fine moduli space. By considering framed U(r)‐instanton bundles as a real structure on framed instanton bundles over P, we show that the bijection between isomorphism classes of framed U(r)‐instanton bundles and isomorphism classes of framed vector bundles over (S, F) due to [5] is actually an isomorphism of moduli spaces.     

Moduli spaces of meromorphic connections and quiver varieties

We describe the moduli spaces of meromorphic connections on trivial holomorphic vector bundles over the Riemann sphere with at most one (unramified) irregular singularity and arbitrary number of simple poles as Nakajima's quiver varieties. This result enables us to solve partially the additive irregular Deligne–Simpson problem.     

On Simple ℤ2-Invariant and Corner Function Germs

Mathematical Notes - V. I. Arnold has classified simple (i.e., having no moduli for the classification) singularities (function germs), and also simple boundary singularities: function germs...     

Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.     

Moduli spaces for principal bundles in arbitrary characteristic

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.     

Some results on the moduli spaces of quiver bundles

This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can be viewed as sheaves of modules over a sheaf of associative algebras and why this view point is useful, e.g., in their deformation theory. Next we explain the main steps of an algebro-geometric construction of their moduli spaces. Finally, we focus on the special case of holomorphic chains over Riemann surfaces, providing some basic links with quiver representation theory. Combined with the analysis of the homological algebra of quiver sheaves and modules, these links provide a criterion for smoothness of the moduli spaces and tools to study their variation with respect to stability.      

Moduli of plane curve singularities with a single characteristic exponent

This paper studies the moduli space corresponding to irreducible germs of plane analytic curve with a single characteristic exponent. We stratify the moduli space corresponding to such germs using an analytical invariant introduced by Zariski. Then, we compute the minimum Tjurina number on each stratum as well as the dimension of the strata.

     

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.     

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.     

A universal complex structure on complete metrizable topological spaces

Let X be a topological space whose topology may be defined by a complete metric d. Taking all such metrics d we define a universal complex structure on X. For this complex structure the sheaf of germs of holomorphic functions on X coincides with the sheaf of germs of continuous functions on X, and hence the theories of topological and holomorphic vector bundles on X are the same.