Journal of the Ramanujan Mathematical Society

Indranil Biswas ¹, D. S. Nagaraj ²

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, IN
2 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, IN

Abstract

Let S be an irreducible smooth projective surface defined over an algebraically closed field k. For a positive integer d, let Hilb^d (S) be the Hilbert scheme parametrizing the zero-dimensional subschemes of S of length d. For a vector bundle E on S, let H(E) → Hilb^d (S) be its Fourier–Mukai transform constructed using the structure sheaf of the universal subscheme of S × Hilb^d (S) as the kernel. We prove that two vector bundles E and F on S are isomorphic if the vector bundles H(E) and H(F) are isomorphic.

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Indranil Biswas ¹, Shane D’Mello ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, IN

Abstract

We prove that a geometrically irreducible smooth projective curve X of genus g>1 defined over ℝ is an M-variety if and only if Pic⁰(X) is an M-variety. A geometrically irreducible smooth projective curve X defined over ℝ is an M-variety if and only if Pic¹(X) is an M-variety; note that there is no condition on the genus.

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Indranil Biswas ¹, A. J. Parameswaran ²

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, IN
2 Kerala School of Mathematics, Kunnamangalam (PO), Kozhikode-673571, Kerala, IN

Abstract

Let X be an irreducible smooth projective curve defined over ℂ. Fix any integer n≥2. There is a tautological hypersurface Δ∈X×Sⁿ(X), where Sⁿ(X) is the symmetric product. Given any vector bundle E over X, let F(E) be the vector bundle on Sⁿ(X) obtained by taking the direct image of the pullback of E to Δ. Let E and F be semi-stable vector bundles over X such that μ(E),μ(F)>n-1. If F(E) is isomorphic to F(F), then we prove that E is isomorphic to F.

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Indranil Biswas ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, IN

Abstract

In [2], with Balaji and Nagaraj we introduced the ramified principal bundles. The aim here is to introduce the Higgs bundles in the ramified context.

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V. Balaji ¹, Indranil Biswas ²

Affiliations
1 Chennai Mathematical Institute, Sipcot IT Park, Siruseri-603103, IN
2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, IN

Abstract

We correct an error in Theorem 1.1 of [1], as well as extend this theorem.

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Indranil Biswas ¹, Joao Pedro Dos Santos ²

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400 005, IN
2 Universit´e de Paris 6, Institut de Mathematiques de Jussieu, 4, Place Jussieu, 75005, Paris, FR

Abstract

Let X be a separably rationally connected smooth projective variety defined over an algebraically closed field K. If E −→ X is a vector bundle satisfying the condition that for every morphism γ : P¹_K −→ X the pull-back γ ∗E is trivial, we prove that E is trivial. If E −→ X is a strongly semistable vector bundle such that c1(E) and c₂(E) are numerically equivalent to zero, we prove that E is trivial. We also show that X does not admit any nontrivial stratified sheaf. These results are also generalized to principal bundles over X.

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Indranil Biswas ¹

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, IN

Abstract

Let S be a locally complete intersection subvariety of ℂℙ^d of pure dimension d − 2 . Serre gave a construction of a rank two vector bundle over ℂℙ^d under the assumption that the determinant of the normal bundle to S extends to ℂℙ^d. We give an explicit description of this rank two vector bundle in the special case where S is the singular locus of a codimension one foliation on a complex manifold M. This foliation is assumed to satisfy certain nondegeneracy conditions. The rank two vector bundle is obtained from certain transversal jet bundles, associated to a foliation, that are constructed here.

AMS (2000) Subject Classification. 13C10, 13B25.

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Vikraman Balaji ¹, Indranil Biswas ²

Affiliations
1 Chennai Mathematical Institute, 92, G.N. Chetty Road, Chennai-600 017, IN
2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay-400005, IN

Abstract

Semistable principal bundles with vanishing Chern classes over abelian varieties and complete homogeneous spaces in arbitrary characteristics are studied. The results of [14] on vector bundles are generalized to the context of principal bundles. If E_G is a semistable principal G–bundle over a complete homogeneous space with vanishing characteristic classes, then E_G is trivial. A semistable principal G–bundle over an abelian variety admits a reduction of structure group to a Borel subgroup of G. In [20] it was proved that a semistable vector bundle with vanishing Chern classes over a projective manifold X with vanishing Chern classes admits a flat connection. We prove that a principal G–bundle over X, where G is a complex reductive group, with vanishing characteristic classes admits a flat connection.

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Indranil Biswas ¹, Christophe Mourougane ²

Affiliations
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road Mumbai-400005, IN
2 Institut de Mathematiques, de Jussieu Boite 247 4, Place Jussieu 75 252, Paris Cedex 05, FR

Abstract

Let X be a smooth projective manifold. Assume that X is Fano with Picard number one. Let J5bea holomorphic vector bundle over X admitting a holomorphic connection. We prove that E admits a flat holomorphic connection.

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