Abstract:
We consider filtered holomorphic vector bundles on a compact Riemann surface X equipped with a holomorphic connection satisfying a certain transversality condition with respect to the filtration. If Q is a stable vector bundle of rank r and degree (1−genus(X))nr, then any holomorphic connection on the jet bundle Jn(Q) satisfies this transversality condition for the natural filtration of Jn(Q) defined by projections to lower-order jets. The vector bundle Jn(Q) admits holomorphic connection. The main result is the construction of a bijective correspondence between the space of all equivalence classes of holomorphic vector bundles on X with a filtration of length n together with a holomorphic connection satisfying the transversality condition and the space of all isomorphism classes of holomorphic differential operators of order n whose symbol is the identity map.

Abstract:
The results of Biswas (2000) are extended to the situation of transversely projective foliations. In particular, it is shown that a transversely holomorphic foliation defined using everywhere locally nondegenerate maps to a projective space ℂℙn, and whose transition functions are given by automorphisms of the projective space, has a canonical transversely projective structure. Such a foliation is also associated with a transversely holomorphic section of N⊗−k for each k∈[3,n

Abstract:
Let $E_G$ be a stable principal $G$--bundle over a compact connected Kaehler manifold, where $G$ is a connected reductive linear algebraic group defined over the complex numbers. Let $H\subset G$ be a complex reductive subgroup which is not necessarily connected, and let $E_H\subset E_G$ be a holomorphic reduction of structure group. We prove that $E_H$ is preserved by the Einstein-Hermitian connection on $E_G$. Using this we show that if $E_H$ is a minimal reductive reduction in the sense that there is no complex reductive proper subgroup of $H$ to which $E_H$ admits a holomorphic reduction of structure group, then $E_H$ is unique in the following sense: For any other minimal reductive reduction $(H', E_{H'})$ of $E_G$, there is some element $g$ of $G$ such that $H'= g^{-1}Hg$ and $E_{H'}= E_Hg$. As an application, we give an affirmative answer to a question of Balaji and Koll\'ar.

Abstract:
We consider all complex projective manifolds X that satisfy at least one of the following three conditions: 1. There exists a pair $(C ,\varphi)$, where $C$ is a compact connected Riemann surface and $\varphi : C\to X$ a holomorphic map, such that the pull back $\varphi^*TX$ is not semistable. 2. The variety $X$ admits an \'etale covering by an abelian variety. 3. The dimension $\dim X \leq 1$. We conjecture that all complex projective manifolds are of the above type, and prove that the following classes are among those that are of the above type. i) All $X$ with a finite fundamental group. ii) All $X$ such that there is a nonconstant morphism from the projective line to $X$. iii) All $X$ such that the canonical line bundle $K_X$ is either positive or negative or $c_1(K_X) \in H^2(X, {\mathbb Q})$ vanishes. iv) All projective surfaces.

Abstract:
Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x_0. Let \varpi(M,x_0) denote the corresponding fundamental group--scheme introduced by Nori. Let E_G be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization \xi on M. We prove that the following three statements are equivalent: The principal G-bundle E_G over M is given by a homomorphism \varpi(M,x_0) --> G. There are integers b > a > 0 such that the principal G-bundle (F^b_M)^*E_G is isomorphic to (F^a_M)^*E_G, where F_M is the absolute Frobenius morphism of M. The principal G-bundle E_G is strongly semistable, degree(c_2(ad(E_G))c_1(\xi)^{d-2}) = 0, where d = \dim M, and degree(c_1(E_G(\chi))c_1(\xi)^{d-1}) = 0 for every character \chi of G, where E_G(\chi) is the line bundle over $M$ associated to $E_G$ for \chi. The equivalence between the first statement and the third statement was proved by S. Subramanian under the extra assumption that dim(M) = 1 and $G$ is semisimple.

Abstract:
Let $M$ be a compact complex manifold equipped with a Gauduchon metric. If $TM$ is holomorphically trivial, and (V, \theta) is a stable SL(r,{\mathbb C})-Higgs bundle on $M$, then we show that $\theta= 0$. We show that the correspondence between Higgs bundles and representations of the fundamental group for a compact Kaehler manifold does not extend to compact Gauduchon manifolds. This is done by applying the above result to G/\Gamma, where $\Gamma$ is a discrete torsionfree cocompact subgroup of a complex semisimple group $G$.

Abstract:
In \cite{BR1}, \cite{BR2}, 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.

Abstract:
Let G/P be a rational homogeneous variety, where P is a parabolic subgroup of a simple and simply connected linear algebraic group G defined over an algebraically closed field of characteristic zero. A homogeneous principal bundle over G/P is semistable (respectively, polystable) if and only if it is equivariantly semistable (respectively, equivariantly polystable). A stable homogeneous principal H-bundle (E_H ,\rho) is equivariantly stable, but the converse is not true in general. If a homogeneous principal H-bundle (E_H ,\rho) is not equivariantly stable but not stable, then E_H admits an action \rho' of G such that the pair (E_H ,\rho') is a homogeneous principal H-bundle which is not equivariantly stable.

Abstract:
In ``Ramified G-bundles as parabolic bundles'' (J. Ramanujan Math. Soc. (2003) Vol. 18) with Balaji and Nagaraj we introduced the ramified principal bundles. The aim here is to define the Higgs bundles in the ramified context.

Abstract:
Let $X$ be a compact connected Riemann surface of genus at least two. The main theorem of arxiv:1010.1488 says that for any positive integer $n \leq 2({\rm genus}(X)-1)$, the symmetric product $S^n(X)$ does not admit any K\"ahler metric satisfying the condition that all the holomorphic bisectional curvatures are nonnegative. Our aim here is to give a very simple and direct proof of this result of B\"okstedt and Rom\~ao.