### A remark on Hodge cycles on moduli spaces of rank 2 bundles over curves.

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We investigate connections, and more generally logarithmic connections, on holomorphic principal bundles over a compact connected Riemann surface.

The moduli space of stable vector bundles over a moving curve is constructed, and on this a generalized Weil-Petersson form is constructed. Using the local Riemann-Roch formula of Bismut-Gillet-Soulé it is shown that the generalized Weil-Petersson form is the curvature of the determinant line bundle, equipped with the Quillen metric, for a vector bundle on the fiber product of the universal moduli space with the universal curve.

We show that the Néron–Severi group of the Prym variety for a degree three unramified Galois covering of a hyperelliptic Riemann surface has a distinguished subgroup of rank three. For the general hyperelliptic curve, the algebra of Hodge cycles on the Prym variety is generated by this group of rank three.

Let X be a compact Riemann surface and associated to each point p-i of a finite subset S of X is a positive integer m-i. Fix an elliptic curve C. To this data we associate a smooth elliptic surface Z fibered over X. The group C acts on Z with X as the quotient. It is shown that the space of all vector bundles over Z equipped with a lift of the action of C is in bijective correspondence with the space of all parabolic bundles over X with parabolic structure over S and the parabolic weights at any...

In [6], orbifold G-bundles on a certain class of elliptic fibrations over a smooth complex projective curve X were related to parabolic G-bundles over X. In this continuation of [6] we define and investigate holomorphic connections on an orbifold G-bundle over an elliptic fibration.

Let X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects: equivalence classes of α-invariant K-connections on X α-invariant gauge classes of K-connections on X, andα-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic Kℂ-bundle Q → X and a K-reduction P of Q (when...

Let $X$ and ${X}^{\prime}$ be compact Riemann surfaces of genus $\ge 3$, and let $G$ and ${G}^{\prime}$ be nonabelian reductive complex groups. If one component ${\mathcal{M}}_{G}^{d}\left(X\right)$ of the coarse moduli space for semistable principal $G$–bundles over $X$ is isomorphic to another component ${\mathcal{M}}_{{G}^{\prime}}^{{d}^{\prime}}\left({X}^{\prime}\right)$, then $X$ is isomorphic to ${X}^{\prime}$.

We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold. This shows that many complex manifolds admit no or few holomorphic Cartan geometries.

Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\phantom{\rule{0.166667em}{0ex}}\subset \phantom{\rule{0.166667em}{0ex}}G$ be a maximal compact subgroup. Let $X$, $Y$ be irreducible smooth complex projective varieties and $f:X\to Y$ an algebraic morphism, such that ${\pi}_{1}\left(Y\right)$ is virtually nilpotent and the homomorphism ${f}_{*}:{\pi}_{1}\left(X\right)\to {\pi}_{1}\left(Y\right)$ is surjective. Define $$\begin{array}{cc}\hfill {\mathcal{R}}^{f}({\pi}_{1}\left(X\right),G)& =\{\rho \in Hom({\pi}_{1}\left(X\right),G)\mid A\circ \rho \phantom{\rule{4pt}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{factors}\phantom{\rule{4.0pt}{0ex}}\text{through}\phantom{\rule{4.0pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{f}_{*}\}\phantom{\rule{0.166667em}{0ex}},\hfill \\ \hfill {\mathcal{R}}^{f}({\pi}_{1}\left(X\right),K)& =\{\rho \in Hom({\pi}_{1}\left(X\right),K)\mid A\circ \rho \phantom{\rule{4pt}{0ex}}\phantom{\rule{4.0pt}{0ex}}\text{factors}\phantom{\rule{4.0pt}{0ex}}\text{through}\phantom{\rule{4.0pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{f}_{*}\}\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$ where $A:G\to GL(Lie(G\left)\right)$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal{R}}^{f}({\pi}_{1}(X,{x}_{0}),\phantom{\rule{0.166667em}{0ex}}G)/\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}/G$ admits a deformation retraction to ${\mathcal{R}}^{f}({\pi}_{1}(X,{x}_{0}),\phantom{\rule{0.166667em}{0ex}}K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements...

Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M C (r; L) be the coarse moduli space of semistable vector bundles E over C of rank r with ∧r E = L. We show that the Brauer group of any desingularization of M C(r; L) is trivial.

Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 2. Let N be a connected component of the moduli space of semistable principal PGLr (ℂ)-bundles over X; it is a normal unirational complex projective variety. We prove that the Brauer group of a desingularization of N is trivial.

We show that a map between complex-analytic manifolds, at least one ofwhich is in the Fujiki class, is a biholomorphism under a natural condition on the second cohomologies. We use this to establish that, with mild restrictions, a certain relation of “domination” introduced by Gromov is in fact a partial order.

Let $X$ be a smooth projective curve defined over an algebraically closed field $k$, and let ${F}_{X}$ denote the absolute Frobenius morphism of $X$ when the characteristic of $k$ is positive. A vector bundle over $X$ is called virtually globally generated if its pull back, by some finite morphism to $X$ from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of $k$ is positive, a vector bundle $E$ over $X$ is virtually globally generated if and only if ${\left({F}_{X}^{m}\right)}^{*}E\phantom{\rule{0.166667em}{0ex}}\cong \phantom{\rule{0.166667em}{0ex}}{E}_{a}\oplus {E}_{f}$ for...

Given a complex manifold M equipped with an action of a group G, and a holomorphic principal H–bundle EH on M, we introduce the notion of a connection on EH along the action of G, which is called a G–connection. We show some relationship between the condition that EH admits a G–equivariant structure and the condition that EH admits a (flat) G–connection. The cases of bundles on homogeneous spaces and smooth toric varieties are discussed.

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