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Indranil Biswas

Publications and source records attributed to Indranil Biswas.

At least 271 records · Page 15Linked to original sources
arXiv2018

The universal connection for principal bundles over homogeneous spaces and twistor space of coadjoint orbits

Given a holomorphic principal bundle $Q\, \longrightarrow\, X$, the universal space of holomorphic connections is a torsor $C_1(Q)$ for $\text{ad} Q \otimes T^*X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic connection. When $X\,=\, G/P$, where $P$ is a parabolic subgroup of a complex simple group $G$, and $Q$ is the frame bundle of an ample line bundle, we show that $C_1(Q)$ may be identified with $G/L$, where $L\, \subset\, P$ is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-Kähler metric on $T^*(G/P)$, recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.

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arXiv2018

Holomorphic Cartan geometries on complex tori

In [DM] it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affimatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type (G,H), with G a complex affine Lie group, on any complex torus is translation invariant.

math.DG↗
arXiv2018

Branched holomorphic Cartan geometries and Calabi-Yau manifolds

We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi-Yau manifolds do not admit branched holomorphic projective structures. The key ingredient of its proof is the following result of independent interest: If E is a holomorphic vector bundle over a compact simply connected Kähler Calabi-Yau manifold, and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle equipped with the trivial connection.

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arXiv2017

Sasakian and parabolic Higgs bundles

Let $M$ be a quasi-regular compact connected Sasakian manifold, and let $N = M/S^1$ be the base projective variety. We establish an equivalence between the class of Sasakian $G$-Higgs bundles over $M$ and the class of parabolic (or equivalently, ramified) $G$-Higgs bundles over the base $N$.

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arXiv2017

Stability Of The Parabolic Poincaré Bundle

Given a compact Riemann surface $X$ and a moduli space $M_α(Λ)$ of parabolic stable bundles on it of fixed determinant of complete parabolic flags, we prove that the Poincaré parabolic bundle on $X\times M_α(Λ)$ is parabolic stable with respect to a natural polarization on $X\times M_α(Λ)$.

math.AG↗
arXiv2017

Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure

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.

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arXiv2017

On the Automorphisms of a Rank One Deligne-Hitchin Moduli Space

Let $X$ be a compact connected Riemann surface of genus $g \geq 2$, and let ${\mathcal M}_{\rm DH}$ be the rank one Deligne-Hitchin moduli space associated to $X$. It is known that ${\mathcal M}_{\rm DH}$ is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on $X$. We investigate the group $\operatorname{Aut}({\mathcal M}_{\rm DH})$ of all holomorphic automorphisms of ${\mathcal M}_{\rm DH}$. The connected component of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ containing the identity automorphism is computed. There is a natural element of $H^2({\mathcal M}_{\rm DH}, {\mathbb Z})$. We also compute the subgroup of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ that fixes this second cohomology class. Since ${\mathcal M}_{\rm DH}$ admits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that ${\mathcal M}_{\rm DH}$ is Moishezon.

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arXiv2017

Equivariant bundles and adapted connections

Given a complex manifold $M$ equipped with a holomorphic action of a connected complex Lie group $G$, and a holomorphic principal $H$--bundle $E_H$ over $X$ equipped with a $G$--connection $h$, we investigate the connections on the principal $H$--bundle $E_H$ that are (strongly) adapted to $h$. Examples are provided by holomorphic principal $H$--bundles equipped with a flat partial connection over a foliated manifold.

math.DG↗
arXiv2017

Towers of torsors over a field

Let $X$ be a projective, connected and smooth scheme defined over an algebraically closed field $k$. In this paper we prove that a tower of finite torsors (i.e., under the action of finite $k$-group schemes) can be dominated by a single finite torsor. Let $G$ be any finite $k$-group scheme and $Y$ any $G$--torsor over $X$ pointed in $y\,\in\, Y(k)$; we define over $Y$, which may not be reduced, in a very natural way the categories of Nori-semistable and essentially finite vector bundles. These categories are proved to be Tannakian. Their Galois $k$-group schemes $π^S(Y,\,y)$ and $π^N(Y,\,y)$, respectively, thus generalize the $S$--fundamental and the Nori fundamental group schemes. The latter still classifies all the finite torsors over $Y$, pointed over $y$. We also prove that they fit in short exact sequences involving $π^S(X,\,x)$ and $π^N(X,\, x)$ respectively, where $x$ is the image of $y$.

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arXiv2017

A vanishing theorem for co-Higgs bundles on the moduli space of bundles

We consider smooth moduli spaces of semistable vector bundles of fixed rank and determinant on a compact Riemann surface $X$ of genus at least $3$. The choice of a Poincaré bundle for such a moduli space $M$ induces an isomorphism between $X$ and a component of the moduli space of semistable sheaves over $M$. We prove that $h^0(M, \text{End}({\mathcal E})\otimes TM)= 1$ for a vector bundle $\mathcal E$ on $M$ coming from this component. Furthermore, there are no nonzero integrable co-Higgs fields on $\mathcal E$.

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arXiv2017

Direct image of parabolic line bundles

Given a vector bundle $E$ on an irreducible projective variety $X$ we give a necessary and sufficient criterion for $E$ to be a direct image of a line bundle under an étale morphism. The criterion in question is the existence of a Cartan subalgebra bundle of the endomorphism bundle $\text{End}(E)$. As a corollary, a criterion is obtained for $E$ to be the direct image of the structure sheaf under an étale morphism. The direct image of a parabolic line bundle under any ramified covering map has a natural parabolic structure. Given a parabolic vector bundle, we give a similar criterion for it to be a direct image of a parabolic line bundle under a ramified covering map.

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arXiv2017

Connections on parahoric torsors over curves

We define parahoric $\cG$--torsors for certain Bruhat--Tits group scheme $\cG$ on a smooth complex projective curve $X$ when the weights are real, and also define connections on them. We prove that a $\cG$--torsor is given by a homomorphism from $π_1(X\setminus D)$ to a maximal compact subgroup of $G$, where $D\, \subset\, X$ is the parabolic divisor, if and only if the torsor is polystable.

math.AG↗
arXiv2017

Embedding algorithms and applications to differential equations

Algorithms for embedding certain types of nilpotent subalgebras in maximal subalgebras of the same type are developed, using methods of real algebraic groups. These algorithms are applied to determine non-conjugate subalgebras of the symmetry algebra of the wave equation, which in turn are used to determine a large class of invariant solutions of the wave equation. The algorithms are also illustrated for the symmetry algebra of a classical system of differential equations considered by Cartan in the context of contact geometry.

math.RT↗
arXiv2017

Logarithmic connections on principal bundles over a Riemann surface

Let $E_G$ be a holomorphic principal $G$-bundle on a compact connected Riemann surface $X$, where $G$ is a connected reductive complex affine algebraic group. Fix a finite subset $D \subset X$, and for each $x\in D$ fix $w_x \in \text{ad}(E_G)_x$. Let $T$ be a maximal torus in the group of all holomorphic automorphisms of $E_G$. We give a necessary and sufficient condition for the existence of a $T$-invariant logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$. We also give a necessary and sufficient condition for the existence of a logarithmic connection on $E_G$ singular over $D$ such that the residue over each $x \in D$ is $w_x$, under the assumption that each $w_x$ is $T$-rigid.

math.AG↗
arXiv2017

Flat connections and cohomology invariants

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 $H^{2r-k}(M,\mathbb{R})$, where $S$ is null-cobordant $(k-1)$-manifold, once a $G$-invariant polynomial $p$ of degree $r$ on $\text{Lie}(G)$ is fixed. For $S=S^{k-1}$, this gives a homomorphism $π_{k-1}(\mathcal{F})\longrightarrow H^{2r-k}(M,\mathbb{R})$. The map is shown to be globally gauge invariant and furthermore it descends to the moduli space of flat connections $\mathcal{F}/\mathrm{Gau}P$, modulo cohomology with integer coefficients. The construction is also adapted to complex manifolds. In this case, one works with the set $\mathcal{F}^{0,2}$ of connections with vanishing $(0,2)$-part of the curvature, and the Dolbeault cohomology. Some examples and applications are presented.

math.DG↗
arXiv2017

A Torelli theorem for moduli spaces of principal bundles on curves defined over $\mathbb R$

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 $X$.

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arXiv2017

Criterion for logarithmic connections with prescribed residues

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 assumption that the residues $A(x)$ are rigid.

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