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### Biswas, Indranil

- Fourier-Mukai Transform of Vector Bundles on Surfaces to Hilbert Scheme

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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

#### Authors

**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

#### Source

Journal of the Ramanujan Mathematical Society, Vol 32, No 1 (2017), Pagination: 43-50#### 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.

- A Criterion for
*M*-Curves

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

#### Authors

**Affiliations**

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

#### Source

Journal of the Ramanujan Mathematical Society, Vol 30, No 4 (2015), Pagination: 403-411#### 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 Pic

^{0}(

*X*) is an

*M*-variety. A geometrically irreducible smooth projective curve

*X*defined over ℝ is an

*M*-variety if and only if Pic

^{1}(

*X*) is an

*M-*variety; note that there is no condition on the genus.

- Vector Bundles on Symmetric Product of a Curve

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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

#### Authors

**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

#### Source

Journal of the Ramanujan Mathematical Society, Vol 26, No 3 (2011), Pagination: 351-355#### Abstract

Let*X*be an irreducible smooth projective curve defined over ℂ. Fix any integer

*n*≥2. There is a tautological hypersurface Δ∈

*X*×

*S*(

^{n}*X*), where

*S*(

^{n}*X*) is the symmetric product. Given any vector bundle

*E*over

*X*, let

*F*(

*E*) be the vector bundle on

*S*(

^{n}*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.*

- Parabolic Principal Higgs Bundles

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

#### Authors

Indranil Biswas

^{1}**Affiliations**

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

#### Source

Journal of the Ramanujan Mathematical Society, Vol 23, No 3 (2008), Pagination: 311–325#### 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.- Principal Bundles on Abelian Varieties With Vanishing Chern Classes

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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

#### Authors

**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

#### Source

Journal of the Ramanujan Mathematical Society, Vol 24, No 2 (2009), Pagination: 191-197#### Abstract

We correct an error in Theorem 1.1 of [1], as well as extend this theorem.- Triviality Criteria for Bundles Over Rationally Connected Varieties

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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

#### Authors

**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

#### Source

Journal of the Ramanujan Mathematical Society, Vol 28, No 4 (2013), Pagination: 423–442#### 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 : P^{1}

_{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 c

_{2}(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.

- Serre’s Construction of Rank Two Vector Bundles and the Transversal Jet Bundles of Certain Codimension One Holomorphic Foliations

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

#### Authors

Indranil Biswas

^{1}**Affiliations**

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

#### Source

Journal of the Ramanujan Mathematical Society, Vol 16, No 1 (2001), Pagination: 1-17#### 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*under the assumption that the determinant of the normal bundle to

^{d}*S*extends to ℂℙ

*. We give an explicit description of this rank two vector bundle in the special case where*

^{d}*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.

- On Principal Bundles with Vanishing Chern Classes

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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

#### Authors

**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

#### Source

Journal of the Ramanujan Mathematical Society, Vol 17, No 3 (2002), Pagination: 187–209#### 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.

- Holomorphic Connection on a Fano Manifold with Picard Number One

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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

#### Authors

**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