arXiv subjects
Publications and source records attributed to Indranil Biswas.
For complex connected, reductive, affine, algebraic groups $G$, we give a Lie-theoretic characterization of the semistability of principal $G$-co-Higgs bundles on the complex projective line $\mathbb{P}^1$ in terms of the simple roots of a Borel subgroup of $G$. We describe a stratification of the moduli space in terms of the Harder-Narasimhan type of the underlying bundle.
Let $φ: Y \rightarrow X$ be a finite surjective morphism between smooth complex projective curves, where $X$ is irreducible but $Y$ need not be so. Let $V_*$ be a parabolic vector bundle on $Y$. We construct a parabolic structure on the direct image $φ_* V$ on $X$, where $V$ is the vector bundle underlying $V_*$. The parabolic vector bundle $φ_* V_*$ on $X$ obtained this way has a ramified torus sub-bundle; it is a torus bundle of $\text{Ad}(φ_* V)$ outside the parabolic divisor for $φ_* V_*$ that satisfies certain conditions at the parabolic points. Conversely, given a parabolic vector bundle $E_*$ on $X$, and a ramified torus sub-bundle $\mathcal T$ for it, we construct a ramified covering $Z$ of $X$ and a parabolic vector bundle $W_*$ on $Z$, such that the parabolic bundle $E_*$ is the direct image of $W_*$. A connection on $V_*$ produces a connection on $φ_* V_*$. The ramified torus sub-bundle for $φ_* V_*$ is preserved by the logarithmic connection on $\text{End}(φ_* V)$ induced by this connection on $φ_* V_*$. If the parabolic vector bundle $E_*$ on $X$ is equipped with a connection $D$ such that the connection on the endomorphism bundle induced by it preserves the ramified torus sub-bundle $\mathcal T$, then we prove that the corresponding parabolic vector bundle $W_*$ on $Z$ has a connection that produces the connection $D$ on the direct image $E_*$.
We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favour for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type result holds, and that there are metrics on stable bundles over surfaces whose Schur forms are positive.
When a complex semisimple group $G$ acts holomorphically on a Kähler manifold $(X,ω)$ such that a maximal compact subgroup $K\subset G$ preserves the symplectic form $ω$, a basic result of symplectic geometry says that the corresponding categorical quotient $X/G$ can be identified with quotient of the zero-set of the moment map by the action of $K$. We extend this to the context of a semisimple group acting on a Sasakian manifold.
Given a real hyperelliptic algebraic curve $X$ with non-empty real part and a real effective divisor $\mc{D}$ arising via pullback from $\mathbb{P}^1$ under the hyperelliptic structure map, we study the real inflection points of the associated complete real linear series $|\mc{D}|$ on $X$. To do so we use Viro's patchworking of real plane curves, recast in the context of some Berkovich spaces studied by M. Jonsson. Our method gives a simpler and more explicit alternative to limit linear series on metrized complexes of curves, as developed by O. Amini and M. Baker, for curves embedded in toric surfaces.
We show that the structure of proper holomorphic maps between the $n$-fold symmetric products, $n\geq 2$, of a pair of non-compact Riemann surfaces $X$ and $Y$, provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of $X$ onto $Y$. This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all $n$-fold symmetric products of a non-compact Riemann surface.
A vector bundle E on a projective variety X is called finite if it satisfies a nontrivial polynomial equation with integral coefficients. A theorem of Nori implies that E is finite if and only if the pullback of E to some finite etale Galois covering of X is trivial. We prove the same statement when X is a compact complex manifold admitting a Gauduchon astheno-Kahler metric.
We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G / Γ$, where $G$ is a complex connected Lie group and $Γ$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_ρ$ associated to any irreducible representation $ρ: Γ\rightarrow \text{GL}(r,{\mathbb C})$. More precisely, we prove that $E_ρ$ is holomorphically isomorphic to a vector bundle of the form $E^{\oplus n}$, where $E$ is a stable vector bundle. All the rational Chern classes of $E$ vanish, in particular, its degree is zero. We deduce a stability result for flat holomorphic vector bundles $E_ρ$ of rank 2 over $G/ Γ$. If an irreducible representation $ρ: Γ\rightarrow \text{GL}(2, \mathbb {C})$ satisfies the conditionmthat the induced homomorphism $Γ\rightarrow {\rm PGL}(2, {\mathbb C})$ does not extend to a homomorphism from $G$, then $E_ρ$ is proved to be stable.
We study the set ${\mathcal P}_S$ consisting of all branched holomorphic projective structures on a compact Riemann surface $X$ of genus $g \geq 1$ and with a fixed branching divisor $S:= \sum_{i=1}^d n_i\cdot x_i$, where $x_i \in X$. Under the hypothesis that $n_i=1$, for all $i$, with $d$ a positive even integer such that $d \neq 2g-2$, we show that ${\mathcal P}_S$ coincides with a subset of the set of all logarithmic connections with singular locus $S$, satisfying certain geometric conditions, on the rank two holomorphic jet bundle $J^1(Q)$, where $Q$ is a fixed holomorphic line bundle on $X$ such that $Q^{\otimes 2}= TX\otimes {\mathcal O}_X(S)$. The space of all logarithmic connections of the above type is an affine space over the vector space $H^0(X, K^{\otimes 2}_X \otimes {\mathcal O}_X(S))$ of dimension $3g-3+d$. We conclude that ${\mathcal P}_S$ is a subset of this affine space that has codimenison $d$ at a generic point.
Let $E_G$ be a $Γ$--equivariant algebraic principal $G$--bundle over a normal complex affine variety $X$ equipped with an action of $Γ$, where $G$ and $Γ$ are complex linear algebraic groups. Suppose $X$ is contractible as a topological $Γ$--space with a dense orbit, and $x_0 \in X$ is a $Γ$--fixed point. We show that if $Γ$ is reductive, then $E_G$ admits a $Γ$--equivariant isomorphism with the product principal $G$--bundle $X \times_ρ E_G(x_0)$, where $ρ\,:\, Γ\, \longrightarrow\, G$ is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal $G$-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal $G$-bundles over any complex toric variety.
Let $M$ be a compact complex manifold of dimension at least three and $Π: M\rightarrow X$ a positive principal elliptic fibration, where $X$ is a compact Kähler orbifold. Fix a preferred Hermitian metric on $M$. In \cite{V}, the third author proved that every stable vector bundle on $M$ is of the form $L\otimes Π^*B_0$, where $B_0$ is a stable vector bundle on $X$, and $L$ is a holomorphic line bundle on $M$. Here we prove that every stable Higgs bundle on $M$ is of the form $(L\otimes Π^*B_0,Π^*Φ_X)$, where $(B_0, Φ_X)$ is a stable Higgs bundle on $X$ and $L$ is a holomorphic line bundle on $M$.
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 $Ω^0_{\HH}$) holomorphic on the punctured surface $S' = S - \{p_1, \cdots, p_n\}$ with exponential singularities at the points $p_1, \cdots, p_n$ of types $[h_1], \cdots, [h_n]$, i.e., near $p_i$ any $f \in \OO_{\HH}$ is of the form $f = ge^{h_i}$ for some germ of meromorphic function $g$ (respectively, any $ω\in Ω^0_{\HH}$ is of the form $ω= αe^{h_i}$ for some germ of meromorphic $1$-form). For any $ω\in Ω^0_{\HH}$ the completion of $S'$ with respect to the flat metric $|ω|$ gives a space $S^* = S' \cup \RR$ obtained by adding a finite set $\RR$ of $\sum_i d_i$ points, and it is known that integration along curves produces a nondegenerate pairing of the relative homology $H_1(S^*, \RR ; \C)$ with the deRham cohomology group defined by $H^1_{dR}(S, \HH) := Ω^0_{\HH}/d\OO_{\HH}$. There is a degree zero line bundle $L_{\HH}$ associated to an exp-algebraic curve, with a natural isomorphism between $Ω^0_{\HH}$ and the space $W_{\HH}$ of meromorphic $L_{\HH}$-valued $1$-forms which are holomorphic on $S'$, so that $H_1(S^*, \RR ; \C)$ maps to a subspace $K_{\HH} \subset W^*_{\HH}$. We show that the exp-algebraic curve $(S, \HH)$ is determined uniquely by the pair $(L_{\HH},\, K_{\HH} \subset W^*_{\HH})$.
We construct a vector bundle $E$ on a smooth complex projective surface $X$ with the property that the restriction of $E$ to any smooth closed curve in $X$ admits an algebraic connection while $E$ does not admit any algebraic connection.
Let $X$ be a compact connected Riemann surface of genus $g\geq 0$, and let ${\rm Sym}^d(X)$, $d \ge 1$, denote the $d$-fold symmetric product of $X$. We show that ${\rm Sym}^d(X)$ admits a Hermitian metric with negative Chern scalar curvature if and only if $g \geq 2$, and positive Chern scalar curvature if and only if $d > g$.
Let $X$ be an irreducible smooth projective curve, defined over an algebraically closed field $k$, of genus at least three and $L$ a line bundle on $X$. Let ${\mathcal M}_X(r,L)$ be the moduli space of stable vector bundles on $X$ of rank $r$ and determinant $L$ with $r\geq 2$. We prove that the Brauer group ${\rm Br}(\mathcal{M}_X(r,L))$ is cyclic of order ${\rm g.c.d.}(r,{\rm degree}(L))$. We also prove that ${\rm Br}(\mathcal{M}_X(r,L))$ is generated by the class of the projective bundle obtained by restricting the universal projective bundle. These results were proved earlier in \cite{BBGN} under the assumption that $k=\mathbb C$.
In this article, we establish a Hitchin-Kobayashi type correspondence for generalised Seiberg-Witten monopole equations on Kahler surfaces. We show that the "stability" criterion we obtain, for the existence of solutions, coincides with that of the usual Seiberg-Witten monopole equations. This enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors.
Given a compact Riemann surface $X$ and a complex reductive Lie group $G$ equipped with real structures, we define antiholomorphic involutions on the moduli space of $G$-Higgs bundles over $X$. We investigate how the various components of the fixed point locus match up, as one passes from $G$ to its Langlands dual $^LG$. Special attention is given to the case $G=SL(2,\C)$ and $^LG=PGL(2,\C)$.
Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension $d$ admits, away from a closed analytic subset of positive codimension, a nonsingular holomorphic foliation of complex codimension $d$ endowed with a transversely flat branched complex projective geometry (equivalently, a ${\mathbb C}P^d$-geometry). We also prove that transversely branched holomorphic Cartan geometries on compact complex projective rationally connected varieties and on compact simply connected Calabi-Yau manifolds are always flat (consequently, they are defined by holomorphic maps into homogeneous spaces).