• INDRANIL BISWAS

Articles written in Proceedings – Mathematical Sciences

• Parabolic ample bundles III: Numerically effective vector bundles

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.

• Stability of Picard bundle over moduli space of stable vector bundles of rank two over a curve

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.

• The determinant bundle on the moduli space of stable triples over a curve

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.

• The Jacobian of a nonorientable Klein 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 Jacobian of a nonorientable Klein surface, II

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

• Torsionfree Sheaves over a Nodal Curve of Arithmetic Genus One

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.

• A Note on the Tangent Bundle of $G/P$

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.

• The Atiyah Bundle and Connections on a Principal Bundle

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.

• On Rationality of Moduli Spaces of Vector Bundles on Real Hirzebruch Surfaces

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.

• Meromorphic connections on vector bundles over curves

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.

• Unitary representations of the fundamental group of orbifolds

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.

• Abelianization of the $F$-divided fundamental group scheme

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.

• Line bundles and flat connections

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.

• $M$-curves and symmetric products

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 $n \geq 2g − 1$.

• Symplectic reduction of Sasakian manifolds

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.

• On stability of tangent bundle of toric varieties

Let $X$ be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle $TX$. In particular, a complete answer is given when $X$ is a Fano toric variety of dimension four with Picard number at most two, complementing the earlier work of Nakagawa (Tohoku. Math. J. 45 (1993) 297--310; 46 (1994) 125--133). We also give an infinite set of examples of Fano toric varieties for which $TX$ is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko (Izv. Akad. Nauk. SSSR Ser. Mat. 53 (1989) 1001--1039, 1135) and developed further by Perling (Math. Nachr. 263/264 (2004) 181--197) and Kool (Moduli spaces of sheaves on toric varieties, Ph.D. thesis (2010) (University of Oxford); Adv. Math. 227 (2011) 1700--1755).

• A normal variety of invariant connexions on Hermitian symmetric spaces

We introduce a class of $G$-invariant connexions on a homogeneous principal bundle $Q$ over a Hermitian symmetric space $M = G/K$. The parameter space carries the structure of normal variety and has a canonical anti-holomorphic involution. The fixed points of the anti-holomorphic involution are precisely the integrable invariant complex structures on $Q$. This normal variety is closely related to quiver varieties and, more generally, to varieties of commuting matrix tuples modulo simultaneous conjugation.

• Chen--Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surface

Let $(X, D)$ be an $m$-pointed compact Riemann surface of genus at least 2. For each $x \in D$, fix full flag and concentrated weight system $\alpha$. Let $P\mathcal{M}_\xi$ denote the moduli space of semi-stable parabolic vector bundles of rank $r$ and determinant $\xi$ over $X$ with weight system $\alpha$, where $r$ is a prime number and $\xi$ is a holomorphic line bundle over $X$ of degree $d$ which is not a multiple of $r$ . We compute the Chen--Ruan cohomology of the orbifold for the action on $P\mathcal{M}_\xi$ of the group of $r$-torsion points in ${\rm Pic}^0(X)$.

• # Proceedings – Mathematical Sciences

Volume 132, 2022
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019