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Indranil Biswas

Publications and source records attributed to Indranil Biswas.

At least 199 records · Page 11Linked to original sources
arXiv2021

Higgs bundles and flat connections over compact Sasakian manifolds

Given a compact Kähler manifold $X$, there is an equivalence of categories between the completely reducible flat vector bundles on $X$ and the polystable Higgs bundles $(E,\, θ)$ on $X$ with $c_1(E)= 0= c_2(E)$ \cite{SimC}, \cite{Cor}, \cite{UY}, \cite{DonI}. We extend this equivalence of categories to the context of compact Sasakian manifolds. We prove that on a compact Sasakian manifold, there is an equivalence between the category of semi-simple flat bundles on it and the category of polystable basic Higgs bundles on it with trivial first and second basic Chern classes. We also prove that any stable basic Higgs bundle over a compact Sasakian manifold admits a basic Hermitian metric that satisfies the Yang--Mills--Higgs equation.

math.DG↗
arXiv2021

Ramified covering maps and stability of pulled back bundles

Let $f:C\rightarrow D$ be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that $f$ is genuinely ramified if ${\mathcal O}_D$ is the maximal semistable subbundle of $f_*{\mathcal O}_C$ (equivalently, the homomorphism of etale fundamental groups is surjective). We prove that the pullback $f^*E\rightarrow C$ is stable for every stable vector bundle $E$ on $D$ if and only if $f$ is genuinely ramified.

math.AG↗
arXiv2021

One-relator Sasakian groups

We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n \geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.

math.AG↗
arXiv2020

A Hodge theoretic projective structure on Riemann surfaces

Given any compact Riemann surface $C$, there is a canonical meromorphic 2--form $\widehatη$ on $C\times C$, with pole of order two on the diagonal $Δ\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic 2--form $\widehatη$ produces a canonical projective structure on $C$. On the other hand the uniformization theorem provides another canonical projective structure on any compact Riemann surface $C$. We prove that these two projective structures differ in general. This is done by comparing the $(0,1)$--component of the differential of the corresponding sections of the moduli space of projective structures over the moduli space of curves. The $(0,1)$--component of the differential of the section corresponding to the projective structure given by the uniformization theorem was computed by Zograf and Takhtadzhyan in \cite{ZT} as the Weil--Petersson Kähler form $ω_{wp}$ on the moduli space of curves. We prove that the $(0,1)$--component of the differential of the section of the moduli space of projective structures corresponding to $\widehatη$ is the pullback of a nonzero constant scalar multiple of the Siegel form, on the moduli space of principally polarized abelian varieties, by the Torelli map.

math.AG↗
arXiv2020

Stability of the Poincaré bundle

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 $d \in π_1(G)$, 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 Poincaré adjoint bundle over $X \times M_G^{d, rs}$, where $M_G^{d, rs}$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.

math.AG↗
arXiv2020

A normal variety of invariant connections on hermitian symmetric spaces

We introduce a class of $G$-invariant connections on a homogeneous principal bundle $Q$ over a hermitian symmetric space $M=G/K$. The parameter space carries the structure of normal variety and has a canonical anti-holomorphic involution. The fixed points of the anti-holomorphic involution are precisely the integrable invariant complex structures on $Q.$ This normal variety is closely related to quiver varieties and, more generally, to varieties of commuting matrix tuples modulo simultaneous conjugation.

math.DG↗
arXiv2020

On stability of tangent bundle of toric varieties

Let $X$ be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle $T{X}$. In particular, a complete answer is given when $X$ is a Fano toric variety of dimension four with Picard number at most two, complementing earlier work of Nakagawa. We also give an infinite set of examples of Fano toric varieties for which $TX$ is unstable; the dimensions of this collection of varieties are unbounded. Our method is based on the equivariant approach initiated by Klyachko and developed further by Perling and Kool.

math.AG↗
arXiv2020

Local topological obstruction for divisors

Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in $H^2(\mathcal{O}_X)$. In this article, we replace $H^2(\mathcal{O}_X)$ by $H^2_D(\mathcal{O}_X)$ and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of $D$ as an effective Cartier divisor of a first order infinitesimal deformations of $X$). We apply this to study the jumping locus of families of linear systems and the Noether-Lefschetz locus. Finally, we give examples of first order deformations $X_t$ of $X$ for which the cohomology class $[D]$ deforms as a Hodge class but $D$ does not lift as an effective Cartier divisor of $X_t$.

math.AG↗
arXiv2020

Tamely ramified torsors and parabolic bundles

Given a variety $X$, and a normal crossings divisor $D\subset X$, we relate, in the case of abelian monodromy, the following two: 1. existence of a $G$-torsor with prescribed ramification, and 2. existence of essentially finite parabolic vector bundles with prescribed weights.

math.AG↗
arXiv2020

Real Higgs pairs and Non-abelian Hodge correspondence on a Klein surface

We introduce real structures on $L$-twisted Higgs pairs over a compact Riemann surface equipped with an anti-holomorphic involution, and prove a Hitchin--Kobayashi correspondence for them. Real $G$-Higgs bundles, where $G$ is a real form of a connected semisimple complex affine algebraic group $G^{\mathbb{C}}$, constitute a particular class of examples of these pairs. The real structure in this case involves a conjugation of $G^{\mathbb{C}}$ commuting with the one defining the real form $G$. We establish a homeomorphism between the moduli space of real $G$-Higgs bundles and the moduli space of compatible representations of the orbifold fundamental group of $X$. Finally, we show how real $G$-Higgs bundles appear naturally as fixed points of certain anti-holomorphic involutions of the moduli space of $G$-Higgs bundles, that are constructed using the real structures on $G^{\mathbb{C}}$ and $X$.

math.DG↗
arXiv2020

Meromorphic connections, determinant line bundles and the Tyurin parametrization

We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the "sheaf of holomorphic connections" (the sheaf of splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic. The equivalences, both holomorphic and symplectic, seem to be quite general, in that they extend to other general families of holomorphic bundles and holomorphic connections, in particular those arising from "Tyurin families" of stable bundles over the surface. These families generalize the Tyurin parametrization of stable vector bundles $E$ over a compact connected Riemann surface, and one can build above them spaces of (equivalence classes of) connections, which are again symplectic. These spaces are also symplectically biholomorphically equivalent to the sheaf of connections for the determinant bundle over the Tyurin family. The last portion of the paper shows how this extends to moduli of framed bundles.

math.AG↗
arXiv2020

Complex Lagrangians in a hyperKaehler manifold and the relative Albanese

Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold $X$, and let $\varpi : \widehat{\mathcal{A}} \rightarrow M$ be the relative Albanese over $M$. We prove that $\widehat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\widehat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\widehat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$.

math.DG↗
arXiv2020

Toric co-Higgs bundles on toric varieties

Starting from the data of a nonsingular complex projective toric variety, we define an associated notion of toric co-Higgs bundle. We provide a Lie-theoretic classification of these objects by studying the interaction between Klyachko's fan filtration and the fiber of the co-Higgs bundle at a closed point in the open orbit of the torus action. This can be interpreted, under certain conditions, as the construction of a coarse moduli scheme of toric co-Higgs bundles of any rank and with any total equivariant Chern class.

math.AG↗
arXiv2020

Holomorphic bundles trivializable by proper surjective holomorphic map

Given a compact complex manifold $M$, we investigate the holomorphic vector bundles $E$ on $M$ such that $φ^* E$ is trivial for some surjective holomorphic map $φ$, to $M$, from some compact complex manifold. We prove that these are exactly those holomorphic vector bundles that admit a flat holomorphic connection with finite monodromy homomorphism. A similar result is proved for holomorphic principal $G$-bundles, where $G$ is a connected reductive complex affine algebraic group.

math.AG↗
arXiv2020

Positivity of vector bundles on homogeneous varieties

We study the following question: Given a vector bundle on a projective variety $X$ such that the restriction of $E$ to every closed curve $C \,\subset\, X$ is ample, under what conditions $E$ is ample? We first consider the case of an abelian variety $X$. If $E$ is a line bundle on $X$, then we answer the question in the affirmative. When $E$ is of higher rank, we show that the answer is affirmative under some conditions on $E$. We then study the case of $X \,=\, G/P$, where $G$ is a reductive complex affine algebraic group, and $P$ is a parabolic subgroup of $G$. In this case, we show that the answer to our question is affirmative if $E$ is $T$--equivariant, where $T\, \subset\, P$ is a fixed maximal torus. Finally, we compute the Seshadri constant for such vector bundles defined on $G/P$.

math.AG↗
arXiv2020

Holomorphic GL(2)-geometry on compact complex manifolds

We study holomorphic GL(2) and SL(2) geometries on compact complex manifolds. We show that a compact Kähler manifold of complex even dimension higher than two admitting a holomorphic GL(2)-geometry is covered by a compact complex torus. We classify compact Kähler-Einstein manifolds and Fano manifolds bearing holomorphic GL(2)-geometries. Among the compact Kähler-Einstein manifolds we prove that the only examples bearing holomorphic GL(2)-geometry are those covered by compact complex tori, the three dimensional quadric and those covered by the three dimensional Lie ball (the non compact dual of the quadric).

math.DG↗