Abstract:
Let X be a smooth projective connected curve of genus $g \ge 2$ and let I be a finite set of points of X. Fix a parabolic structure on I for rank r vector bundles on X. Let $M^{par}$ denote the moduli space of parabolic semistable bundles and let $L^{par}$ denote the parabolic determinant bundle. In this paper we show that the n-th tensor power line bundle ${L^{par}}^n$ on the moduli space $M^{par}$ is globally generated, as soon as the integer n is such that $n \ge [\frac{r^2}{4}]$. In order to get this bound, we construct a parabolic analogue of the Quot scheme and extend the result of Popa and Roth on the estimate of its dimension.

Abstract:
This article deals with two topics: the first, which has a general character, is a variation formula for the the determinant line bundle in non-K\"ahlerian geometry. This formula, which is a consequence of the non-K\"ahlerian version of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the variation of the determinant line bundle corresponding to a perturbation of a Fourier-Mukai kernel ${\cal E}$ on a product $B\times X$ by a unitary flat line bundle on the fiber $X$. When this fiber is a complex surface and ${\cal E}$ is a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant. The second topic concerns a geometric application of our variation formula, namely we will study compact complex subspaces of the moduli spaces of stable bundles considered in our program for proving existence of curves on minimal class VII surfaces. Such a moduli space comes with a distinguished point $a=[{\cal A}]$ corresponding to the canonical extension ${\cal A}$ of $X$. The compact subspaces $Y\subset {\cal M}^\st$ containing this distinguished point play an important role in our program. We will prove a non-existence result: there exists no compact complex subspace of positive dimension $Y\subset {\cal M}^\st$ containing $a$ with an open neighborhood $a\in Y_a\subset Y$ such that $Y_a\setminus\{a\}$ consists only of non-filtrable bundles. In other words, within any compact complex subspace of positive dimension $Y\subset {\cal M}^\st$ containing $a$, the point $a$ can be approached by filtrable bundles. Specializing to the case $b_2=2$ we obtain a new way to complete the proof of a theorem in a previous article: any minimal class VII surface with $b_2=2$ has a cycle of curves. Applications to class VII surfaces with higher $b_2$ will be be discussed in a forthcoming article.

Abstract:
In this article we extend the proof given by Biswas and Gomez of a Torelli theorem for the moduli space of Higgs bundles with fixed determinant, to the parabolic situation.

Abstract:
The moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities associated to the quasi-parabolic structure is equal to one. It follows that if rank and degree are coprime, the moduli space of vector bundles is stably rational, and the bound obtained on the level is strong enough to conclude rationality in many cases.

Abstract:
Let ${\mathcal P}{\mathcal M}^\alpha_s$ be a moduli space of stable parabolic vector bundles of rank $n \geq 2$ and fixed determinant of degree $d$ over a compact connected Riemann surface $X$ of genus $g(X) \geq 2$. If $g(X) = 2$, then we assume that $n > 2$. Let $m$ denote the greatest common divisor of $d$, $n$ and the dimensions of all the successive quotients of the quasi-parabolic filtrations. We prove that the cohomological Brauer group ${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)$ is isomorphic to the cyclic group ${\mathbb Z}/ m{\mathbb Z}$. We also show that ${\rm Br}({\mathcal P}{\mathcal M}^\alpha_s)$ is generated by the Brauer class of the projective bundle over ${\mathcal P}{\mathcal M}^\alpha_s$ obtained by restricting the universal projective bundle over $X\times {\mathcal P}{\mathcal M}^\alpha_s$. We also prove that there is a universal vector bundle over $X\times {\mathcal P}{\mathcal M}^\alpha_s$ if and only if $m=1$.

Abstract:
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 interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kaehler 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 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.

Abstract:
Parabolic triples of the form $(E_*,\theta,\sigma)$ are considered, where $(E_*,\theta)$ is a parabolic Higgs bundle on a given compact Riemann surface $X$ with parabolic structure on a fixed divisor $S$, and $\sigma$ is a nonzero section of the underlying vector bundle. Sending such a triple to the Higgs bundle $(E_*,\theta)$ a map from the moduli space of stable parabolic triples to the moduli space of stable parabolic Higgs bundles is obtained. The pull back, by this map, of the symplectic form on the moduli space of stable parabolic Higgs bundles will be denoted by $\text{d}\Omega'$. On the other hand, there is a map from the moduli space of stable parabolic triples to a Hilbert scheme $\text{Hilb}^\delta(Z)$, where $Z$ denotes the total space of the line bundle $K_X\otimes{\mathcal O}_X(S)$, that sends a triple $(E_*,\theta,\sigma)$ to the divisor defined by the section $\sigma$ on the spectral curve corresponding to the parabolic Higgs bundle $(E_*,\theta)$. Using this map and a meromorphic one--form on $\text{Hilb}^\delta(Z)$, a natural two--form on the moduli space of stable parabolic triples is constructed. It is shown here that this form coincides with the above mentioned form $\text{d}\Omega'$.

Abstract:
We study natural families of d-bar 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.

Abstract:
Given an $n$-tuple of positive real numbers $\alpha$ we consider the hyperpolygon space $X(\alpha)$, the hyperk\"{a}hler quotient analogue to the K\"ahler moduli space of polygons in $\mathbb{R}^3$. We prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable, rank-$2$, holomorphically trivial parabolic Higgs bundles over $\mathbb{C} \mathbb{P}^1$ with fixed determinant and trace-free Higgs field. This isomorphism allows us to prove that hyperpolygon spaces $X(\alpha)$ undergo an elementary transformation in the sense of Mukai as $\alpha$ crosses a wall in the space of its admissible values. We describe the changes in the core of $X(\alpha)$ as a result of this transformation as well as the changes in the nilpotent cone of the corresponding moduli spaces of parabolic Higgs bundles. Moreover, we study the intersection rings of the core components of $X(\alpha)$. In particular, we find generators of these rings, prove a recursion relation in $n$ for their intersection numbers and use it to obtain explicit formulas for the computation of these numbers. Using our isomorphism, we obtain similar formulas for each connected component of the nilpotent cone of the corresponding moduli spaces of parabolic Higgs bundles thus determining their intersection rings. As a final application of our isomorphism we describe the cohomology ring structure of these moduli spaces of parabolic Higgs bundles and of the components of their nilpotent cone.

Abstract:
The moduli space of stable parabolic vector bundles of parabolic degree 0 over the Riemann sphere is considered. The vector bundle analog of the Klein's Hauptmodul is defined and the regular locus, a subset of bundles with minimal Birkhoff-Grothendieck decomposition and generic Bruhat type of the constant term at $\infty$, is introduced. For the restriction of the natural Kaehler metric to the regular locus a potential is constructed as the value of the regularized WZNW functional evaluated on singular Hermitian metrics in the corresponding vector bundles. It is shown that this potential is an antiderivative of a (1,0)-form on the regular locus, associated with a solution of the Riemann-Hilbert problem.