In this continuation of [Bi2] and [BN], we define numerically effective vector bundles in the parabolic category. Some properties of the usual numerically effective vector bundles are shown to be valid in the more general context of numerically effective parabolic vector bundles.

Answering a question of [BV] it is proved that the Picard bundle on the moduli space of stable vector bundles of rank two, on a Riemann surface of genus at least three, with fixed determinant of odd degree is stable.

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,ϕ), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, andϕ: E₂→ E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.

Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surfaceY is constructed. The Jacobian is identified with the dual of the space of all harmonic real one-forms onY quotiented by the torsion-free part of the first integral homology ofY. Denote byX the double cover ofY given by orientation. The Jacobian ofY is identified with the space of all degree zero holomorphic line bundlesL overX with the property thatL is isomorphic to σ*/-L, where σ is the involution ofX.

The aim here is to continue the investigation in [1] of Jacobians of a Klein surface and also to correct an error in [1].

We classify all isomorphism classes of stable torsionfree sheaves on an irreducible nodal curve of arithmetic genus one defined over $\mathbb{C}$. Let 𝑋 be a nodal curve of arithmetic genus one defined over $\mathbb{R}$, with exactly one node, such that 𝑋 does not have any real points apart from the node. We classify all isomorphism classes of stable real algebraic torsionfree sheaves over 𝑋 of even rank. We also classify all isomorphism classes of real algebraic torsionfree sheaves over 𝑋 of rank one.

Let 𝑃 be a parabolic subgroup of a complex simple linear algebraic group 𝐺. We prove that the tangent bundle $T(G/P)$ is stable.

Let 𝑀 be a $C^\infty$ manifold and 𝐺 a Lie a group. Let $E_G$ be a $C^\infty$ principal 𝐺-bundle over 𝑀. There is a fiber bundle $\mathcal{C}(E_G)$ over 𝑀 whose smooth sections correspond to the connections on $E_G$. The pull back of $E_G$ to $\mathcal{C}(E_G)$ has a tautological connection. We investigate the curvature of this tautological connection.

Let 𝑋 be a real form of a Hirzebruch surface. Let $M_H(r,c_1,c_2)$ be the moduli space of vector bundles on 𝑋. Under some numerical conditions on $r,c_1$ and $c_2$, we identify those $M_H(r,c_1,c_2)$ that are rational.

We give a criterion for filtered vector bundles over curves to admit a filtration preserving meromorphic connection that induces a given meromorphic connection on the corresponding graded bundle.

Let $X$ be a smooth complex projective variety of dimension $n$ and $\mathcal{L}$ an ample line bundle on it. There is a well known bijective correspondence between the isomorphism classes of polystable vector bundles $E$ on $X$ with $c_{1}(E) = 0 = c_{2}(E) \cdot c_{1} \mathcal (L)^{n−2}$ and the equivalence classes of unitary representations of $\pi_{1}(X)$. We show that this bijective correspondence extends to smooth orbifolds.

Let ($X , x_0$) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for ($X , x_0$) produces a homomorphism from the abelianization of the $F$-divided fundamental group scheme of $X$ to the $F$-divided fundamental group of the Albanese variety of $X$. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.

We prove that there are cocompact lattices $\Gamma$ in $\rm SL(2,\mathbb C)$ with the property that there are holomorphic line bundles $L$ on $\rm SL(2,\mathbb C)/ \Gamma$ with $c_{1}(L) = 0$ such that $L$ does not admit any unitary flat connection.

Let $(X , \sigma)$ be a geometrically irreducible smooth projective $M$-curve of genus $g$ defined over the field of real numbers.We prove that the $n$-th symmetric product of $(X , \sigma)$ is an $M$-variety for $n$ = 2 ,3 and $n \geq 2g − 1$.

When a complex semisimple group $G$ acts holomorphically on a K\"ahler manifold $(X,\, \omega)$ such that a maximal compact subgroup $K\, \subset\, G$ preserves the symplectic form $\omega$, a basic result of symplectic geometry says that the corresponding categorical quotient $X/G$ can be identified with the quotient of the zero-set of the moment map by the action of $K$. We extend this to the context of a semisimple group acting on a Sasakian manifold.