Indranil Biswas
Articles written in Proceedings – Mathematical Sciences
Volume 109 Issue 1 February 1999 pp 41-46
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.
Volume 111 Issue 3 August 2001 pp 263-269
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.
Volume 112 Issue 3 August 2002 pp 367-382
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 (E_{1}, E_{2},
Volume 113 Issue 2 May 2003 pp 139-152
The Jacobian of a nonorientable Klein surface
Pablo Arés-Gastesi Indranil Biswas
Using divisors, an analog of the Jacobian for a compact connected nonorientable Klein surface
Volume 116 Issue 1 February 2006 pp 51-58
The Jacobian of a nonorientable Klein surface, II
Pablo Arés-Gastesi Indranil Biswas
The aim here is to continue the investigation in [1] of Jacobians of a Klein surface and also to correct an error in [1].
Volume 118 Issue 1 February 2008 pp 81-98
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.
Volume 120 Issue 1 February 2010 pp 69-71
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.
Volume 120 Issue 3 June 2010 pp 299-316
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.
Volume 123 Issue 2 May 2013 pp 213-223
On Rationality of Moduli Spaces of Vector Bundles on Real Hirzebruch Surfaces
Indranil Biswas Ronnie Sebastian
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.
Volume 124 Issue 4 November 2014 pp 487-496
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.
Volume 126 Issue 4 October 2016 pp 557-575 Research Article
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.
Volume 127 Issue 2 April 2017 pp 281-287 Research Article
Abelianization of the $F$-divided fundamental group scheme
INDRANIL BISWAS JOAO PEDRO P DOS SANTOS
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.
Volume 127 Issue 3 June 2017 pp 547-549 Research Article
Line bundles and flat connections
INDRANIL BISWAS GEORG SCHUMACHER
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.
Volume 127 Issue 4 September 2017 pp 615-624 Research Article
$M$-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 $n \geq 2g − 1$.
Volume 129 Issue 4 September 2019 Article ID 0062 Research Article
Symplectic reduction of Sasakian manifolds
INDRANIL BISWAS GEORG SCHUMACHER
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.
Volume 131 All articles Published: 22 October 2021 Article ID 0036 Article
On stability of tangent bundle of toric varieties
INDRANIL BISWAS ARIJIT DEY OZHAN GENC MAINAK PODDAR
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 (
Volume 131 All articles Published: 23 October 2021 Article ID 0038 Article
A normal variety of invariant connexions on Hermitian symmetric spaces
INDRANIL BISWAS HARALD UPMEIER
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.
Volume 132 All articles Published: 17 May 2022 Article ID 0031 Article
Chen--Ruan cohomology and moduli spaces of parabolic bundles over a Riemann surface
INDRANIL BISWAS PRADEEP DAS ANOOP SINGH
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 ${\rm Pic}^0(X)$.
Volume 132, 2022
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