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by Indranil Biswas
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Let $G$ be a connected reductive complex affine algebraic group and $K\subset G$ a maximal compact subgroup. Let $M$ be a compact complex torus equipped with a flat K\"ahler structure and $(E_G ,\theta)$ a polystable Higgs $G$-bundle on $M$. Take any $C^\infty$ reduction of structure group $E_K \subset E_G$ to the subgroup $K$ that solves the Yang--Mills equation for $(E_G ,\theta)$. We prove that the principal $G$-bundle $E_G$ is polystable and the above reduction $E_K$ solves the...
Topics: Mathematics, Differential Geometry
Source: http://arxiv.org/abs/1411.2882
Arxiv.org
by Indranil Biswas
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A new construction of a universal connection was given in \cite{BHS}. The main aim here is to explain this construction. A theorem of Atiyah and Weil says that a holomorphic vector bundle $E$ over a compact Riemann surface admits a holomorphic connection if and only if the degree of every direct summand of $E$ is degree. In \cite{AB}, this criterion was generalized to principal bundles on compact Riemann surfaces. This criterion for principal bundles is also explained.
Topics: Differential Geometry, Mathematics
Source: http://arxiv.org/abs/1608.02400
Arxiv.org
by Indranil Biswas
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Let $X$ be a smooth complex elliptic curve and $G$ a connected reductive affine algebraic group defined over $\mathbb C$. Let ${\mathcal M}_X(G)$ denote the moduli space of topologically trivial algebraic $G$--connections on $X$, that is, pairs of the form $(E_G\, , D)$, where $E_G$ is a topologically trivial algebraic principal $G$--bundle on $X$, and $D$ is an algebraic connection on $E_G$. We prove that ${\mathcal M}_X(G)$ does not admit any nonconstant algebraic function while being...
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1504.01821
Arxiv.org
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Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant...
Topics: Mathematics, Differential Geometry, Algebraic Geometry
Source: http://arxiv.org/abs/1404.0458
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by Indranil Biswas
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Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via biholomorphisms. We give an explicit description of the isomorphism classes of $G$-equivariant almost holomorphic hermitian principal bundles on $N$. Under the assumption that there is a central subgroup $Z= \text{U}(1)$ of $K(S)$ that acts on $\text{Lie}(N)$ as...
Topics: Mathematics, Complex Variables, Differential Geometry
Source: http://arxiv.org/abs/1502.05121
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by Indranil Biswas; Harish Seshadri
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Let $S^n(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^n(X)$ admits a K\"ahler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n < d$. Let ${\mathcal Q}_X(r,n)$ be the Quot scheme parametrizing the torsion quotients of ${\mathcal O}^{\oplus r}_X$ of degree $n$. If $g \geq 2$ and $n \leq 2g-2$, we prove that ${\mathcal Q}_X(r,n)$ does not admit a K\"ahler...
Topics: Mathematics, Differential Geometry
Source: http://arxiv.org/abs/1401.7408
Arxiv.org
by Indranil Biswas; Shane D'Mello
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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 $n\geq 2g -1$.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1603.00234
Arxiv.org
by Marco Antei; Indranil Biswas
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Let $k$ be an algebraically closed field. Chambert-Loir proved that the \'etale fundamental group of a normal rationally chain connected variety over $k$ is finite. We prove that the fundamental group scheme of a normal rationally chain connected variety over $k$ is finite and \'etale. In particular, the fundamental group scheme of a Fano variety is finite and \'etale.
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1410.1166
Arxiv.org
by Indranil Biswas; Florent Schaffhauser
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Given a geometrically irreducible smooth projective curve of genus 1 defined over the field of real numbers, and a pair of integers r and d, we determine the isomorphism class of the moduli space of semi-stable vector bundles of rank r and degree d on the curve. When r and d are coprime, we describe the topology of the real locus and give a modular interpretation of its points. We also study, for arbitrary rank and degree, the moduli space of indecomposable vector bundles of rank r and degree...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1410.6845
Arxiv.org
by Indranil Biswas; Viktoria Heu
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We construct a pair (E ,F), where E is a holomorphic vector bundle over a compact Riemann surface and F a holomorphic subbundle of E, such that both F and E/F admit holomorphic connections, but E does not.
Topics: Complex Variables, Mathematics
Source: http://arxiv.org/abs/1409.3178
Arxiv.org
by Indranil Biswas; Kingshook Biswas
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An exp-algebraic curve consists of a compact Riemann surface $S$ together with $n$ equivalence classes of germs of meromorphic functions modulo germs of holomorphic functions, $\HH = \{ [h_1], \cdots, [h_n] \}$, with poles of orders $d_1, \cdots, d_n \geq 1$ at points $p_1, \cdots, p_n$. This data determines a space of functions $\OO_{\HH}$ (respectively, a space of $1$-forms $\Omega^0_{\HH}$) holomorphic on the punctured surface $S' = S - \{p_1, \cdots, p_n\}$ with exponential singularities at...
Topics: Complex Variables, Mathematics
Source: http://arxiv.org/abs/1606.06449
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by Indranil Biswas; Niels Borne
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Given an algebraic stack, we compare its Nori fundamental group with that of its coarse moduli space. We also study conditions under which the stack can be uniformized by an algebraic space.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1502.07023
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Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$--bundles on $X$ determine uniquely the isomorphism class of $X$.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1704.04318
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by Indranil Biswas; Carlos Florentino
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Let G be a connected complex reductive affine algebraic group, and let K be a maximal compact subgroup. Let X be a compact connected K\"ahler manifold whose fundamental group Gamma is virtually nilpotent. We prove that the character variety Hom(Gamma, G)/G admits a natural strong deformation retraction to the subset Hom(Gamma, K)/K. The natural action of C^* on the moduli space of G-Higgs bundles over X extends to an action of C. This produces the deformation retraction.
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1405.0610
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by Indranil Biswas; Harish Seshadri
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Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$, and let ${\mathcal O}_X$ denote the sheaf of holomorphic functions on $X$. Fix positive integers $r$ and $d$ and let ${\mathcal Q}(r,d)$ be the Quot scheme parametrizing all torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. We prove that ${\mathcal Q}(r,d)$ does not admit a K\"ahler metric whose holomorphic bisectional curvatures are all nonnegative.
Topics: Differential Geometry, Mathematics
Source: http://arxiv.org/abs/1503.08530
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by Indranil Biswas; Mahan Mj
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Given an almost complex structure on a subbundle of the cotangent bundle, we prove a Castelnuovo--de Franchis type theorem for it.
Topics: Mathematics, Complex Variables
Source: http://arxiv.org/abs/1502.06205
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by Indranil Biswas; Carlos Florentino
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Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\subset G$ be a maximal compact subgroup. Let $X , Y$ be irreducible smooth complex projective varieties and $f: X \rightarrow Y$ an algebraic morphism, such that $\pi_1(Y)$ is virtually nilpotent and the homomorphism $f_* : \pi_1(X) \rightarrow\pi_1(Y)$ is surjective. Define $$ {\mathcal R }^f(\pi_1(X),\, G)\,=\, \{\rho\, \in\, \text{Hom}(\pi_1(X),\, G)\, \mid\, A\circ\rho \ \text{ factors through...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1507.04568
Arxiv.org
by Indranil Biswas; Vamsi Pingali
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We prove that structured vector bundles whose holonomies lie in GL(N,C), SO(N,C), or Sp(2N,C) have structured inverses. This generalizes a theorem of Simons and Sullivan.
Topics: Mathematics, Differential Geometry
Source: http://arxiv.org/abs/1502.00071
Arxiv.org
by Indranil Biswas; Sukhendu Mehrotra
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Given a compact connected Riemann surface $X$ of genus $g \geq 2$, and integers $r\geq 2$, $d_p > 0$ and $d_z > 0$, in \cite{BDHW}, a generalized quot scheme ${\mathcal Q}_X(r,d_p,d_z)$ was introduced. Our aim here is to compute the holomorphic automorphism group of ${\mathcal Q}_X(r,d_p,d_z)$. It is shown that the connected component of $\text{Aut}( {\mathcal Q}_X(r,d_p,d_z))$ containing the identity automorphism is $\text{PGL}(r,{\mathbb C})$. As an application of it, we prove that if...
Topics: Algebraic Geometry, Mathematical Physics, Mathematics
Source: http://arxiv.org/abs/1601.04576
Arxiv.org
by Indranil Biswas; Graeme Wilkin
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We study complex Lagrangian submanifolds of a compact hyper-K\"ahler manifold and prove two results: (a) that an involution of a hyper-K\"ahler manifold which is antiholomorphic with respect to one complex structure and which acts non-trivially on the corresponding symplectic form always has a fixed point locus which is complex Lagrangian with respect to one of the other complex structures, and (b) there exist Lagrangian submanifolds which are complex with respect to one complex...
Topics: Mathematics, Mathematical Physics, Differential Geometry
Source: http://arxiv.org/abs/1410.6616
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by Indranil Biswas; Jacques Hurtubise
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We consider monopoles with singularities of Dirac type on quasiregular Sasakian three-folds fibering over a compact Riemann surface $\Sigma$, for example the Hopf fibration $S^3\longrightarrow S^2$. We show that these correspond to holomorphic objects on $\Sigma$, which we call twisted bundle triples. These are somewhat similar to Murray's bundle gerbes. A spectral curve construction allows us to classify these structures, and, conjecturally, monopoles.
Topics: Mathematics, Mathematical Physics, Algebraic Geometry
Source: http://arxiv.org/abs/1412.4050
Arxiv.org
by Indranil Biswas; Ananyo Dan
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Given a family $\pi:\mc{X} \rightarrow B$ of smooth projective varieties, a closed fiber $\mc{X}_o$ and an invertible sheaf $\mc{L}$ on $\mc{X}_o$, we compare the Hodge locus in $B$ corresponding to the Hodge class $c_1(\mc{L})$ with the locus of points $b\,\in\, B$ such that $\mc{L}$ deforms to an invertible sheaf $\mc{L}_b$ on $\mc{X}_b$ with at least $h^0(\mc{L})$--dimensional space of global sections (it is a Brill-Noether type locus associated to $\mc{L}$). We finally give an application...
Topics: Complex Variables, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1609.00997
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by Indranil Biswas; Sean Lawton
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Let S be a surface of genus g with n points removed, G a connected Lie group, and X(G) the moduli space of representations of the fundamental group of S into G. We compute the fundamental group of X(G) when n>0 and G is a real or complex reductive algebraic group, or a compact Lie group; and when n=0 and G=GL(m,C), SL(m,C), U(m), or SU(m).
Topics: Mathematics, Algebraic Topology, Algebraic Geometry
Source: http://arxiv.org/abs/1405.3580
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by Indranil Biswas; Jacques Hurtubise
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One can associate to many of the well known algebraically integrable systems of Jacobians (generalized Hitchin systems, Sklyanin) a ruled surface which encodes much of its geometry. If one looks at the classification of such surfaces, there is one case of a ruled surface that does not seem to be covered. This is the case of projective bundle associated to the first jet bundle of a topologically nontrivial line bundle. We give the integrable system corresponding to this surface; it turns out to...
Topics: Symplectic Geometry, Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1410.1138
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by Indranil Biswas; Georg Schumacher
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Let $X\rightarrow S$ be a smooth projective surjective morphism of relative dimension $n$, where $X$ and $S$ are integral schemes over $\mathbb C$. Let $L\rightarrow X$ be a relatively very ample line bundle. For every sufficiently large positive integer $m$, there is a canonical isomorphism of the Deligne pairing $\langle L ,\cdots , L\rangle\rightarrow S$ with the determinant line bundle ${\rm Det}((L- {\mathcal O}_{X})^{\otimes (n+1)}\otimes L^{\otimes m})$ \cite{PRS}. If we fix a hermitian...
Topics: Differential Geometry, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1501.02539
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by Indranil Biswas; Mahan Mj
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We initiate the study of the asymptotic topology of groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the $H_1$-semistability conjecture of Geoghegan for holomorphically convex groups. In view of a theorem of Eyssidieux, Katzarkov, Pantev and Ramachandran \cite{ekpr}, this implies that linear projective groups satisfy the $H_1$-semistability...
Topics: Group Theory, Mathematics, Geometric Topology
Source: http://arxiv.org/abs/1403.2029
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We characterize the quasiprojective groups that appear as fundamental groups of compact $3$-manifolds (with or without boundary). We also characterize all closed $3$-manifolds that admit good complexifications. These answer questions of Friedl--Suciu, \cite{fs}, and Totaro \cite{tot}
Topics: Geometric Topology, Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1402.6418
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by Indranil Biswas; Arjun Paul
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Let $X$ be a connected complex manifold equipped with a holomorphic action of a complex Lie group $G$. We investigate conditions under which a principal bundle on $X$ admits a $G$--equivariance structure.
Topics: Complex Variables, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1611.08854
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by Indranil Biswas; Sorin Dumitrescu
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Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be K\"ahler. We prove that holomorphic geometric structures of affine type on compact Calabi-Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In particular, if the geometric structure is rigid in Gromov's sense, then the fundamental group of the manifold must be infinite. We also prove that compact complex...
Topics: Complex Variables, Differential Geometry, Mathematics
Source: http://arxiv.org/abs/1602.04677
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We investigate stable holomorphic vector bundles on a compact complex K\"ahler manifold and more generally on an orbifold that is equipped with a K\"ahler structure. We use the existence of Hermite-Einstein connections in this set-up and construct a generalized Weil-Petersson form on the moduli space of stable vector bundles with fixed determinant bundle. We show that the Weil-Petersson form extends as a (semi-)positive closed current for degenerating families that are restrictions of...
Topics: Mathematics, Algebraic Geometry, Complex Variables
Source: http://arxiv.org/abs/1509.00304
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We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is described in a systematic way.
Topics: Complex Variables, Differential Geometry, Representation Theory, Mathematics
Source: http://arxiv.org/abs/1601.02859
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by Indranil Biswas; Harish Seshadri
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Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal Q}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of ${\mathcal Q}_X(r,d)$ is not nef. Equivalently, ${\mathcal Q}_X(r,d)$ does not admit any K\"ahler...
Topics: Differential Geometry, Algebraic Geometry, Mathematical Physics, Mathematics
Source: http://arxiv.org/abs/1703.07753
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by Indranil Biswas; Mahan Mj
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We extend the Donaldson-Corlette-Hitchin-Simpson correspondence between Higgs bundles and flat connections on compact K\"ahler manifolds to compact quasi-regular Sasakian manifolds. A particular consequence is the translation of restrictions on K\"ahler groups proved using the Donaldson-Corlette-Hitchin-Simpson correspondence to fundamental groups of compact Sasakian manifolds.
Topics: Complex Variables, Differential Geometry, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1607.07351
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by Usha N. Bhosle; Indranil Biswas
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We study moduli spaces $M_X(r,c_1,c_2)$ parametrizing slope semistable vector bundles of rank $r$ and fixed Chern classes $c_1, c_2$ on a ruled surface whose base is a rational nodal curve. We show that under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space $M_X(r,c_1,c_2)$ is rational as a variety defined over...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1509.03384
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by Indranil Biswas; Marco Castrillón López
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The main goal of this article is to construct some geometric invariants for the topology of the set $\mathcal{F}$ of flat connections on a principal $G$-bundle $P\,\longrightarrow\, M$. Although the characteristic classes of principal bundles are trivial when $\mathcal{F}\neq \emptyset$, their classical Chern-Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps $S\longrightarrow \mathcal{F}$ to the cohomology group $H^{2r-k}(M,\mathbb{R})$,...
Topics: Differential Geometry, Mathematics
Source: http://arxiv.org/abs/1704.05414
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by Indranil Biswas; D. S. Nagaraj
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Let $S$ be an irreducible smooth projective surface defined over an algebraically closed field $k$. For a positive integer $d$, let ${\rm Hilb}^d(S)$ be the Hilbert scheme parametrizing the zero-dimensional subschemes of $S$ of length $d$. For a vector bundle $E$ on $S$, let ${\mathcal H}(E)\, \longrightarrow\, {\rm Hilb}^d(S)$ be its Fourier--Mukai transform constructed using the structure sheaf of the universal subscheme of $S\times {\rm Hilb}^d(S)$ as the kernel. We prove that two vector...
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1605.06229
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by Indranil Biswas; Tomas L. Gomez
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We show that all the automorphisms of the symmetric product Sym^d(X), d>2g-2, of a smooth projective curve X of genus g>2 are induced by automorphisms of X.
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1506.01500
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by Indranil Biswas; Sanjay Kumar Singh
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Let $C$ be an irreducible smooth projective curve defined over an algebraically closed field. We prove that the symmetric product ${\rm Sym}^d(C)$ has the diagonal property for all $d \geq 1$. For any positive integers $n$ and $r$, let ${\mathcal Q}_{{\mathcal O}^{\oplus n}_C}(nr)$ be the Quot scheme parametrizing all the torsion quotients of ${\mathcal O}^{\oplus n}_C$ of degree $nr$. We prove that ${\mathcal Q}_{{\mathcal O}^{\oplus n}_C}(nr)$ has the weak point property.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1502.07626
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by Indranil Biswas; Nuno M. Romão
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Gauged linear sigma-models at critical coupling on Riemann surfaces yield self-dual field theories, their classical vacua being described by the vortex equations. For local models with structure group ${\rm U}(r)$, we give a description of the vortex moduli spaces in terms of a fibration over symmetric products of the base surface $\Sigma$, which we assume to be compact. Then we show that all these fibrations induce isomorphisms of fundamental groups. A consequence is that all the moduli spaces...
Topics: Mathematics, Mathematical Physics
Source: http://arxiv.org/abs/1503.00526
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by Indranil Biswas; Oscar García-Prada
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We study anti-holomorphic involutions of the moduli space of principal $G$-Higgs bundles over a compact Riemann surface $X$, where $G$ is a complex semisimple Lie group. These involutions are defined by fixing anti-holomorphic involutions on both $X$ and $G$. We analyze the fixed point locus in the moduli space and their relation with representations of the orbifold fundamental group of $X$ equipped with the anti-holomorphic involution. We also study the relation with branes. This generalizes...
Topics: Mathematics, Differential Geometry, Algebraic Geometry
Source: http://arxiv.org/abs/1401.7236
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by Indranil Biswas; Niels Leth Gammelgaard
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We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.
Topics: Mathematics, Differential Geometry, Geometric Topology
Source: http://arxiv.org/abs/1410.6440
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by Indranil Biswas; Arijit Dey; Mainak Poddar
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Let $M$ be a smooth complex projective toric variety equipped with an action of a torus $T$, such that the complement $D$ of the open $T$--orbit in $M$ is a simple normal crossing divisor. Let $G$ be a complex reductive affine algebraic group. We prove that an algebraic principal $G$--bundle $E_G\to M$ admits a $T$--equivariant structure if and only if $E_G$ admits a logarithmic connection singular over $D$. If $E_H\to M$ is a $T$-equivariant algebraic principal $H$--bundle, where $H$ is any...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1507.02415
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by Indranil Biswas; Viktoria Heu; Jacques Hurtubise
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Let $G$ be a reductive affine algebraic group defined over $\mathbb C$, and let $\nabla_0$ be a meromorphic $G$-connection on a holomorphic $G$-bundle $E_0$, over a smooth complex curve $X_0$, with polar locus $P_0 \subset X_0$. We assume that $\nabla_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. We consider the universal isomonodromic deformation $(E_t\to X_t, \nabla_t, P_t)_{t\in \mathcal{T}}$ of $(E_0\to X_0, \nabla_0, P_0)$, where...
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1608.00780
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by Indranil Biswas; Viktoria Heu; Jacques Hurtubise
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For the universal isomonodromic deformation of an irreducible logarithmic rank two connection over a smooth complex projective curve of genus at least two, consider the family of holomorphic vector bundles over curves underlying this universal deformation. In a previous work we proved that the vector bundle corresponding to a general parameter of this family is stable. Here we prove that the vector bundle corresponding to a general parameter is in fact very stable (it does not admit any nonzero...
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1703.07203
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by Indranil Biswas; Ugo Bruzzo; Sudarshan Gurjar
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Relying on a notion of "numerical effectiveness" for Higgs bundles, we show that the category of "numerically flat" Higgs vector bundles on a smooth projective variety $X$ is a Tannakian category. We introduce the associated group scheme, that we call the "Higgs fundamental group scheme of $X$," and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1607.07207
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by Indranil Biswas; Arijit Dey; Mainak Poddar
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We classify holomorphic as well as algebraic torus equivariant principal $G$-bundles over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1510.04014
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by Indranil Biswas; Ritwik Mukherjee; Varun Thakre
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We obtain a formula for the number of genus two curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This is done by extending the symplectic approach of Aleksey Zinger. This enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves on the surface.
Topics: Symplectic Geometry, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1511.04900
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by Gautam Bharali; Indranil Biswas; Georg Schumacher
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Let $X$ and $Y$ be compact connected complex manifolds of the same dimension with $b_2(X)= b_2(Y)$. We prove that any surjective holomorphic map of degree one from $X$ to $Y$ is a biholomorphism. A version of this was established by the first two authors, but under an extra assumption that $\dim H^1(X {\mathcal O}_X)\,=\,\dim H^1(Y {\mathcal O}_Y)$. We show that this condition is actually automatically satisfied.
Topics: Complex Variables, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1610.06286
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by Indranil Biswas; Ritwik Mukherjee; Varun Thakre
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We obtain a formula for the number of genus one curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves.
Topics: Symplectic Geometry, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1509.08284
Arxiv.org
by Indranil Biswas; Viktoria Heu; Jacques Hurtubise
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Let X_0 be a compact connected Riemann surface of genus g with D_0\subset X_0 an ordered subset of cardinality n, and let E_G be a holomorphic principal G-bundle on X_0, where G is a complex reductive affine algebraic group, that admits a logarithmic connection \nabla_0 with polar divisor D_0. Let (\cal{E}_G, \nabla) be the universal isomonodromic deformation of (E_G,\nabla_0) over the universal Teichm\"uller curve (\cal{X}, \cal{D})\rightarrow {Teich}_{g,n}, where {Teich}_{g,n} is the...
Topics: Mathematics, Algebraic Geometry, Complex Variables
Source: http://arxiv.org/abs/1505.05327
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by Indranil Biswas; Ananyo Dan; Arjun Paul
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A theorem of Weil and Atiyah says that a holomorphic vector bundle $E$ on a compact Riemann surface $X$ admits a holomorphic connection if and only if the degree of every direct summand of $E$ is zero. Fix a finite subset $S$ of $X$, and fix an endomorphism $A(x) \in \text{End}(E_x)$ for every $x \in S$. It is natural to ask when there is a logarithmic connection on $E$ singular over $S$ with residue $A(x)$ at every $x \in S$. We give a necessary and sufficient condition for it under the...
Topics: Algebraic Geometry, Complex Variables, Mathematics
Source: http://arxiv.org/abs/1703.09864
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by Reynir Axelsson; Indranil Biswas; Georg Schumacher
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The classical Hurwitz spaces, that parameterize compact Riemann surfaces equipped with covering maps to ${\mathbb P}_1$ of fixed numerical type with simple branch points, are extensively studied in the literature. We apply deformation theory, and present a study of the K\"ahler structure of the Hurwitz spaces, which reflects the variation of the complex structure of the Riemann surface as well as the variation of the meromorphic map. We introduce a generalized Weil-Petersson K\"ahler...
Topics: Mathematics, Algebraic Geometry, Complex Variables
Source: http://arxiv.org/abs/1502.02384
Arxiv.org
by Indranil Biswas; Amit Hogadi; A. J. Parameswaran
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Let $M$ be an irreducible smooth projective variety, defined over an algebraically closed field, equipped with an action of a connected reductive affine algebraic group $G$, and let ${\mathcal L}$ be a $G$--equivariant very ample line bundle on $M$. Assume that the GIT quotient $M/\!\!/G$ is a nonempty set. We prove that the homomorphism of algebraic fundamental groups $\pi_1(M)\, \longrightarrow\, \pi_1(M/\!\!/G)$, induced by the rational map $M\, \longrightarrow\, M/\!\!/G$, is an...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1410.5156
Arxiv.org
by Indranil Biswas; Tomás L. Gómez; Norbert Hoffmann
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Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $\mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d \in \pi_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X \times \mathcal{M}^d_G$ is stable with respect to any polarization on $X \times \mathcal{M}^d_G$. A similar result is proved for the Poincar\'e adjoint bundle over $X...
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1701.04649
Arxiv.org
by Indranil Biswas; Oscar Garcia-Prada; Jacques Hurtubise
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We consider stable and semistable principal bundles over a smooth projective real algebraic curve, equipped with a real or pseudo-real structure in the sense of Atiyah. After fixing suitable topological invariants, one can build a suitable gauge theory, and show that the resulting moduli spaces of pseudo-real bundles are connected. This in turn allows one to describe the various fixed point varieties on the complex moduli spaces under the action of the real involutions on the curve and the...
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1502.00563
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by Indranil Biswas; Pierre-Emmanuel Chaput; Christophe Mourougane
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It is known that the cotangent bundle Omega_Y of an irreducible Hermitian symmetric space Y of compact type is stable. Except for a few obvious exceptions, we show that if $X \subset Y$ is a complete intersection such that Pic(Y) \to Pic(X) is surjective, then the restriction Omega_{Y|X} is stable. We then address some cases where the Picard group increases by restriction.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1504.03853
Arxiv.org
by Indranil Biswas; João Pedro P. dos Santos
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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.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1601.04955
Arxiv.org
by Indranil Biswas; Swarnava Mukhopadhyay; Arjun Paul
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Let $X$ be a compact connected Riemann surface of genus at least two, and let ${G}$ be a connected semisimple affine algebraic group defined over $\mathbb C$. For any $\delta \in \pi_1({G})$, we prove that the moduli space of semistable principal ${G}$--bundles over $X$ of topological type $\delta$ is simply connected. In contrast, the fundamental group of the moduli stack of principal ${G}$--bundles over $X$ of topological type $\delta$ is shown to be isomorphic to $H^1(X, \pi_1({G}))$.
Topics: Algebraic Geometry, Geometric Topology, Algebraic Topology, Mathematics
Source: http://arxiv.org/abs/1609.06436
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by Indranil Biswas; Sean Lawton; Daniel Ramras
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We compute the fundamental group of moduli spaces of Lie group valued representations of surface and torus groups.
Topics: Mathematics, Algebraic Topology, Representation Theory, Algebraic Geometry
Source: http://arxiv.org/abs/1412.4389
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We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight system is generic. When the genus is at least two, using this result we also prove a Torelli theorem for the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise, Integrable systems and algebraic surfaces, Duke...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1411.5040
Arxiv.org
by Indranil Biswas; Sean Lawton; Daniel Ramras
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Using the wonderful compactification of a semisimple adjoint affine algebraic group G defined over an algebraically closed field k of arbitrary characteristic, we construct a natural compactification Y of the G-character variety of any finitely generated group F. When F is a free group, we show that this compactification is always simply connected with respect to the \'etale fundamental group, and when k=C it is also topologically simply connected. For other groups F, we describe conditions for...
Topics: Algebraic Geometry, Symplectic Geometry, Representation Theory, Mathematics
Source: http://arxiv.org/abs/1703.04431
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by Indranil Biswas; Pralay Chatterjee; Chandan Maity
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The second de Rham cohomology groups of nilpotent orbits in non-compact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in non-compact classical simple Lie algebras are computed; they are proven to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general...
Topics: Group Theory, Representation Theory, Algebraic Topology, Mathematics
Source: http://arxiv.org/abs/1611.08369
Arxiv.org
by Indranil Biswas; Saikat Chatterjee; Rukmini Dey
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Given a compact symplectic manifold $M$, with integral symplectic form, we prequantize a certain class of functions on the path space for $M$. The functions in question are induced by functions on $M$. We apply our construction to study the symplectic structure on the solution space of Klein-Gordon equation.
Topics: Quantum Physics, Mathematics, Mathematical Physics, Differential Geometry
Source: http://arxiv.org/abs/1411.5716
Arxiv.org
by David Baraglia; Indranil Biswas; Laura P. Schaposnik
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We compute the automorphism groups of the Dolbeault, de Rham and Betti moduli spaces for the multiplicative group ${\mathbb C}^*$ associated to a compact connected Riemann surface.
Topics: Differential Geometry, Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1508.06587
Arxiv.org
by Indranil Biswas; Pierre-Emmanuel Chaput; Christophe Mourougane
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Let $G$ be an almost simple simply-connected affine algebraic group over an algebraically closed field $k$ of characteristic $p > 0$. If $G$ has type $B_n$, $C_n$ or $F_4$, we assume that $p > 2$, and if $G$ has type $G_2$, we assume that $p > 3$. Let $P \subset G$ be a parabolic subgroup. We prove that the tangent bundle of $G/P$ is Frobenius stable with respect to the anticanonical polarization on $G/P$.
Topics: Mathematics, Algebraic Geometry, Representation Theory
Source: http://arxiv.org/abs/1507.03026
Arxiv.org
by Indranil Biswas; R. V. Gurjar; Sagar U. Kolte
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We consider the Zariski-Lipman Conjecture on free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several classes of affine and projective surfaces.
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1403.5613
Arxiv.org
by Indranil Biswas; S. Senthamarai Kannan; D. S. Nagaraj
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Let $X$ be the wonderful compactification of a complex symmetric space $G/H$ of minimal rank. For a point $x\,\in\, G$, denote by $Z$ be the closure of $BxH/H$ in $X$, where $B$ is a Borel subgroup of $G$. The universal cover of $G$ is denoted by $\widetilde{G}$. Given a $\widetilde{G}$ equivariant vector bundle $E$ on $X,$ we prove that $E$ is nef (respectively, ample) if and only if its restriction to $Z$ is nef (respectively, ample). Similarly, $E$ is trivial if and only if its restriction...
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1501.02540
Arxiv.org
by Indranil Biswas; S. Senthamarai Kannan; D. S. Nagaraj
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Let $T$ be a maximal torus of ${\rm PSL}(n, \mathbb C)$. For $n\,\geq\, 4$, we construct a smooth compactification of ${\rm PSL}(n, \mathbb C)/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{{\rm PSL}(n, \mathbb C)}$ for a suitable choice of $T$--linearized ample line bundle on $\overline{{\rm PSL}(n, \mathbb C)}$. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of ${\rm...
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1501.04031
Arxiv.org
by Indranil Biswas; S. Senthamarai Kannan; D. S. Nagaraj
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Let $G$ be a simple algebraic group of adjoint type over $\mathbb C$, and let $M$ be the wonderful compactification of a symmetric space $G/H$. Take a $\widetilde G$--equivariant principal $R$--bundle $E$ on $M$, where $R$ is a complex reductive algebraic group and $\widetilde G$ is the universal cover of $G$. If the action of the isotropy group $\widetilde H$ on the fiber of $E$ at the identity coset is irreducible, then we prove that $E$ is polystable with respect to any polarization on $M$....
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1501.02541
Arxiv.org
by David Baraglia; Indranil Biswas; Laura P. Schaposnik
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Given a compact Riemann surface $X$ and a semisimple affine algebraic group $G$ defined over $\mathbb C$, there are moduli spaces of Higgs bundles and of connections associated to $(X,\, G)$. We compute the Brauer group of the smooth locus of these varieties.
Topics: Complex Variables, Algebraic Geometry, Geometric Topology, Mathematics
Source: http://arxiv.org/abs/1609.00454
If $L$ is a semisimple Lie algebra of vector fields on R^N with a split Cartan subalgebra C, then it is proved that the dimension of the generic orbit of C coincides with the dimension of C. As a consequence one obtains a local canonical form of L in terms of exponentials of coordinate functions and vector fields that are independent of these coordinates -- for a suitable choice of coordinates. This result is used to classify semisimple algebras of vector fields on R^3 and to determine all...
Topics: Representation Theory, Classical Analysis and ODEs, Mathematics
Source: http://arxiv.org/abs/1612.08535
Arxiv.org
by Indranil Biswas; Niels Leth Gammelgaard; Marina Logares
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Let $X$ be a compact connected Riemann surface of genus at least two. Let $M_H(r,d)$ denote the moduli space of semistable Higgs bundles on $X$ of rank $r$ and degree $d$. We prove that the compact complex Bohr-Sommerfeld Lagrangians of $M_H(r,d)$ are precisely the irreducible components of the nilpotent cone in $M_H(r,d)$. This generalizes to Higgs $G$-bundles and also to the parabolic Higgs bundles.
Topics: Symplectic Geometry, Mathematics, Mathematical Physics, Algebraic Geometry
Source: http://arxiv.org/abs/1409.6814
Arxiv.org
by Indranil Biswas; Mahan Mj; A. J. Parameswaran
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The purpose of this paper is to produce restrictions on fundamental groups of manifolds admitting good complexifications by proving the following Cheeger-Gromoll type splitting theorem: Any closed manifold $M$ admitting a good complexification has a finite-sheeted regular covering $M_1$ such that $M_1$ admits a fiber bundle structure with base $(S^1)^k$ and fiber $N$ that admits a good complexification and also has zero virtual first Betti number. We give several applications to manifolds of...
Topics: Group Theory, Algebraic Geometry, Geometric Topology, Mathematics
Source: http://arxiv.org/abs/1503.08006
Arxiv.org
by Indranil Biswas; Ajneet Dhillon; Jacques Hurtubise; Richard Wentworth
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Let X be a compact connected Riemann surface. Fix a positive integer r and two nonnegative integers d_p and d_z. Consider all pairs of the form (F, f), where F is a holomorphic vector bundle on X of rank r and degree d_z-d_p, and f : {\mathcal O}^{\oplus r}_X \rightarrow F is a meromorphic homomorphism which an isomorphism outside a finite subset of X and has pole (respectively, zero) of total degree d_p (respectively, d_z). Two such pairs $(F_1, f_1) and $(F_2, f_2) are called isomorphic if...
Topics: Mathematics, Mathematical Physics, Algebraic Geometry
Source: http://arxiv.org/abs/1410.1182
Arxiv.org
by Indranil Biswas; Subramaniam Senthamarai Kannan; Donihakalu Shankar Nagaraj
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Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ of adjoint type defined over $\mathbb C.$ Let ${\overline T}\subset \overline G$ be the closure of a maximal torus $T\subset G.$ We prove that the group of all automorphisms of the variety $\overline T$ is the semi-direct product $N_G(T)\rtimes D,$ where $N_G(T)$ is the normalizer of $T$ in $G$ and $D$ is the group of all automorphisms of the Dynkin diagram, if $G\not= {\rm PSL}(2,\mathbb{C})$. Note that...
Topics: Algebraic Geometry, Representation Theory, Mathematics
Source: http://arxiv.org/abs/1702.08364
Arxiv.org
by Indranil Biswas; S. Senthamarai Kannan; D. S. Nagaraj
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Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ defined over $\mathbb C$ such that its center is trivial and $G\not= {\rm PSL}(2,\mathbb{C})$. Take a maximal torus $T \subset G$, and denote by $\overline T$ its closure in $\overline G$. We prove that $T$ coincides with the connected component, containing the identity element, of the group of automorphisms of the variety $\overline T$.
Topics: Group Theory, Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1506.09011
Arxiv.org
by Indranil Biswas; Thomas Koberda; Mahan Mj; Ramanujan Santharoubane
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Let $(S,\, \ast)$ be a closed oriented surface with a marked point, let $G$ be a fixed group, and let $\rho\colon\pi_1(S) \longrightarrow G$ be a representation such that the orbit of $\rho$ under the action of the mapping class group $Mod(S,\, \ast)$ is finite. We prove that the image of $\rho$ is finite. A similar result holds if $\pi_1(S)$ is replaced by the free group $F_n$ on $n\geq 2$ generators and where $Mod(S,\, \ast)$ is replaced by $Aut(F_n)$. We thus resolve a well-known question of...
Topics: Group Theory, Geometric Topology, Mathematics
Source: http://arxiv.org/abs/1702.03622
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by Indranil Biswas; Shane D'Mello; Ritwik Mukherjee; Vamsi Pingali
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We obtain an explicit formula for the number of rational cuspidal curves of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as an Euler class computation on the moduli space of curves. A topological method is employed in computing the contribution of the degenerate locus to this Euler class.
Topics: Mathematics, Algebraic Geometry, Symplectic Geometry
Source: http://arxiv.org/abs/1509.06300
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by Indranil Biswas; Ajneet Dhillon; Jacques Hurtubise; Richard A. Wentworth
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We construct a symplectic analog of the Quot scheme that parametrizes the torsion quotients of a trivial vector bundle over a compact Riemann surface. Some of its properties are investigated.
Topics: Mathematics, Algebraic Geometry
Source: http://arxiv.org/abs/1509.03952
Arxiv.org
by Anilatmaja Aryasomayajula; Indranil Biswas; Archana S. Morye; Tathagata Sengupta
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Let $X$ be a compact connected Riemann surface of genus $g$, with $g \geq 2$. For each $d
Topics: Differential Geometry, Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1608.02207
Arxiv.org
by Hassan Azad; Indranil Biswas; C. S. Rajan; Shehryar Sikander
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Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompact type and $\Gamma \,\subset\, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact K\"ahler manifold and $\rho\, :\, \pi_1(X, x_0)\,\longrightarrow\, \Gamma$ a homomorphism such that the adjoint action of $\rho(\pi_1(X, x_0))$ on $\text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $\rho$ a harmonic map $X\, \longrightarrow\, K\backslash G/\Gamma$. We give a criterion for this...
Topics: Complex Variables, Differential Geometry, Mathematics
Source: http://arxiv.org/abs/1603.02387
Arxiv.org
by Indranil Biswas; Luis Angel Calvo; Emilio Franco; Oscar García-Prada
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We present a systematic study of involutions on the moduli space of $G$-Higgs bundles over an elliptic curve $X$, where $G$ is complex reductive affine algebraic group. The fixed point loci in the moduli space of $G$-Higgs bundles on $X$, and in the moduli space of representations of the fundamental group of $X$ into $G$, are described. This leads to an explicit description of the moduli spaces of pseudo-real $G$-Higgs bundles over $X$.
Topics: Algebraic Geometry, Mathematics
Source: http://arxiv.org/abs/1612.08364
We study quadrature methods for solving Volterra integral equations of the first kind with smooth kernels under the presence of noise in the right-hand sides, with the quadrature methods being generated by linear multistep methods. The regularizing properties of an a priori choice of the step size are analyzed, with the smoothness of the involved functions carefully taken into consideration. The balancing principle as an adaptive choice of the step size is also studied. It is considered in a...
Topics: Numerical Analysis, Mathematics
Source: http://arxiv.org/abs/1604.08703
This paper deals with Lavrentiev regularization for solving linear ill-posed problems, mostly with respect to accretive operators on Hilbert spaces. We present converse and saturation results which are an important part in regularization theory. As a byproduct we obtain a new result on the quasi-optimality of a posteriori parameter choices. Results in this paper are formulated in Banach spaces whenever possible.
Topics: Numerical Analysis, Mathematics
Source: http://arxiv.org/abs/1607.04879
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by Giovanni Bazzoni; Indranil Biswas; Marisa Fernández; Vicente Muñoz; Aleksy Tralle
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We prove the formality and the evenness of odd-degree Betti numbers for compact K\"ahler orbifolds, by adapting the classical proofs for K\"ahler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any K\"ahler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers $b_1=0$ and $b_2\,> 1$.
Topics: Symplectic Geometry, Differential Geometry, Algebraic Topology, Mathematics
Source: http://arxiv.org/abs/1605.03024
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by Plato; Kyriakos Zambas
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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Δ
Topics: State, The -- Early works to 1800, JC, Political science -- Early works to 1800
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by Plato; Kyriakos Zambas
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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Ε
Topics: JC, State, The -- Early works to 1800, Political science -- Early works to 1800
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by Kyriakos Zambas; Plato
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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος B
Topics: Political science -- Early works to 1800, JC, State, The -- Early works to 1800
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by Plato; Kyriakos Zambas
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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Α
Topics: Political science -- Early works to 1800, JC, State, The -- Early works to 1800
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by Plato; Kyriakos Zambas
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Book from Project Gutenberg: Νόμοι και Επινομίς, Τόμος Γ
Topics: JC, Political science -- Early works to 1800, State, The -- Early works to 1800
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by I. N. (Ioannes N.) Grypares; Plato
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Book from Project Gutenberg: Πολιτεία, Τόμος 2
Topics: Utopias -- Early works to 1800, Justice -- Early works to 1800, Political science -- Early works to...
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by I. N. (Ioannes N.) Grypares; Plato
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Book from Project Gutenberg: Πολιτεία, Τόμος 3
Topics: JC, Political science -- Early works to 1800, Utopias -- Early works to 1800, Justice -- Early...
Source: https://www.gutenberg.org/ebooks/39524
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by Plato; I. N. (Ioannes N.) Grypares
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Book from Project Gutenberg: Πολιτεία, Τόμος 4
Topics: Utopias -- Early works to 1800, JC, Justice -- Early works to 1800, Political science -- Early...
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by Plato; I. N. (Ioannes N.) Grypares
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Book from Project Gutenberg: Πολιτεία, Τόμος 1
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