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Indranil Biswas

Publications and source records attributed to Indranil Biswas.

At least 325 records · Page 18Linked to original sources
arXiv2015

Isomonodromic deformations of logarithmic connections and stability

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 Teichmüller curve (\cal{X}, \cal{D})\rightarrow {Teich}_{g,n}, where {Teich}_{g,n} is the Teichmüller space for genus g Riemann surfaces with n-marked points. We prove the following: Assume that g>1 and n= 0. Then there is a closed complex analytic subset \cal{Y} \subset {Teich}_{(g,n)}, of codimension at least $g$, such that for any t\in {Teich}_{(g,n)} \setminus \mathcal{Y}, the principal G-bundle \cal{E}_G\vert_{{\cal X}_t} is semistable, where {\cal X}_t is the compact Riemann surface over $t$. Assume that g>0, and if g= 1, then n >0. Also, assume that the monodromy representation for \nabla_0 does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset $\cal{Y}' \subset {Teich}_{(g,n)}, of codimension at least g, such that for any t\in {Teich}_{(g,n)} \setminus \cal{Y}', the principal G-bundle $\cal{E}_G\vert_{{\cal X}_t}$ is semistable. Assume that g>1. Assume that the monodromy representation for \nabla_0 does not factor through some proper parabolic subgroup of G. Then there is a closed complex analytic subset \cal{Y}" \subset {Teich}_{(g,n)}, of codimension at least g-1, such that for any t\in {Teich}_{(g,n)} \setminus \cal{Y}', the principal G-bundle \cal{E}_G\vert_{{\cal X}_t} is stable.

math.AG↗
arXiv2015

A classification of equivariant principal bundles over nonsingular toric varieties

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.

math.AG↗
arXiv2015

The Period-Index Problem of the Canonical Gerbe of Symplectic and Orthogonal Bundles

We consider regularly stable parabolic symplectic and orthogonal bundles over an irreducible smooth projective curve over an algebraically closed field of characteristic zero. The morphism from the moduli stack of such bundles to its coarse moduli space is a $μ_2$-gerbe. We study the period and index of this gerbe, and solve the corresponding period-index problem.

math.AG↗
arXiv2015

Pseudo-real principal $G$-bundles over a real curve

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 structure group.

math.AG↗
arXiv2015

Rational cuspidal curves on del-Pezzo surfaces

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.

math.AG↗
arXiv2015

A symplectic analog of the Quot scheme

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.

math.AG↗
arXiv2015

Moduli spaces of vector bundles on a singular rational ruled surface

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 $\mathbb R$.

math.AG↗
arXiv2015

The Nori fundamental gerbe of tame stacks

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.

math.AG↗
arXiv2015

Stability of the tangent bundle of G/P in positive characteristics

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 $G/P$.

math.AG↗
arXiv2015

Higgs bundles and representation spaces associated to morphisms

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 $π_1(Y)$ is virtually nilpotent and the homomorphism $f_* : π_1(X) \rightarrowπ_1(Y)$ is surjective. Define $$ {\mathcal R }^f(π_1(X),\, G)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, G)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, , $$ $$ {\mathcal R }^f(π_1(X),\, K)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, K)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, , $$ where $A: G \rightarrow \text{GL}(\text{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal R }^f(π_1(X, x_0), G)/\!\!/G$ admits a deformation retraction to ${\mathcal R }^f(π_1(X, x_0),\, K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements in $G$ admits a deformation retraction to the space of conjugacy classes of $n$ almost commuting elements in $K$.

math.AG↗
arXiv2015

Equivariant principal bundles and logarithmic connections on toric varieties

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 complex affine algebraic group, then $E_H$ in fact has a canonical integrable logarithmic connection singular over $D$.

math.AG↗
arXiv2015

Automorphisms of $\overline{T}$

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 $\overline T$.

math.AG↗
arXiv2015

Yang-Mills-Higgs connections on Calabi-Yau manifolds

Let $X$ be a compact connected Kähler--Einstein manifold with $c_1(TX)\, \geq\, 0$. If there is a semistable Higgs vector bundle $(E\,,θ)$ on $X$ with $θ\,\not=\,0$, then we show that $c_1(TX)=0$, any $X$ satisfying this condition is called a Calabi--Yau manifold, and it admits a Ricci--flat Kähler form \cite{Ya}. Let $(E\,,θ)$ be a polystable Higgs vector bundle on a compact Ricci--flat Kähler manifold $X$. Let $h$ be an Hermitian structure on $E$ satisfying the Yang--Mills--Higgs equation for $(E\,,θ)$. We prove that $h$ also satisfies the Yang--Mills--Higgs equation for $(E\,,0)$. A similar result is proved for Hermitian structures on principal Higgs bundles on $X$ satisfying the Yang--Mills--Higgs equation.

math.AG↗
arXiv2015

On the fundamental group scheme of rationally chain connected varieties

Let $k$ be an algebraically closed field. Chambert-Loir proved that the étale 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 étale. In particular, the fundamental group scheme of a Fano variety is finite and étale.

math.AG↗
arXiv2015

Inverses of structured vector bundles

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.

math.DG↗
arXiv2015

Brauer Groups of Quot Schemes

Let $X$ be an irreducible smooth complex projective curve. Let ${\mathcal Q}(r,d)$ be the Quot scheme parametrizing all coherent subsheaves of ${\mathcal O}^{\oplus r}_X$ of rank $r$ and degree $-d$. There are natural morphisms ${\mathcal Q}(r,d) \longrightarrow \text{Sym}^d(X)$ and $\text{Sym}^d(X) \longrightarrow \text{Pic}^d(X)$. We prove that both these morphisms induce isomorphism of Brauer groups if $d \geq 2$. Consequently, the Brauer group of ${\mathcal Q}(r,d)$ is identified with the Brauer group of $\text{Pic}^d(X)$ if $d \geq 2$.

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