arXiv subjects

Indranil Biswas

Publications and source records attributed to Indranil Biswas.

490 records · Page 28Linked to original sources

arXiv1996

Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the Universal Commensurability Teichmüller Space

There exists on each Teichmüller space $T_g$ (comprising compact Riemann surfaces of genus $g$), a natural sequence of determinant (of cohomology) line bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''. There is a remarkable connection, (Belavin-Knizhnik), between the Mumford isomorphisms and the existence of the Polyakov string measure on the Teichmüller space. This suggests the question of finding a genus-independent formulation of these bundles and their isomorphisms. In this paper we combine a Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q$ bundles to construct such an universal version. Our universal objects exist over the universal space, $T_\infty$, which is the direct limit of the $T_g$ as the genus varies over the tower of all unbranched coverings of any base surface. The bundles and the connecting isomorphisms are equivariant with respect to the natural action of the universal commensurability modular group.

alg-geom↗

arXiv1996

Principle bundles admitting a holomorphic structure

Let $M$ be a compact connected Kähler manifold and let ${\E}_{l-1}$ be the smallest term in the Harder-Narasimhan filtration of its tangent bundle. Let $G$ be an affine algebraic reductive group over $\C$. We prove the following result: If $M$ satisfies the condition that $°(T/{\E}_{l-1}) \geq 0$, then a holomorphic principal $G$-bundle $P$ on $M$ admitting a compatible holomorphic connection is semistable. Moreover, if $°(T/{\E}_{l-1}) >0$, then such a bundle $P$ actually admits a compatible flat $G$-connection.

alg-geom↗

arXiv1995

Parabolic Higgs bundles and Teichmüller spaces for punctured surfaces

In this paper we study the relation between parabolic Higgs bundles and irreducible representations of the fundamental group of punctured Riemann surfaces established by Simpson. We generalize a result of Hitchin, identifying those parabolic Higgs bundles that correspond to Fuchsian representations. We also study the Higgs bundles that give representations whose image is contained, after conjugation, in SL($k,\Bbb R$). We compute the real dimension of one of the components of this space of representations, which in the absence of punctures is the generalized Teichmüller space introduced by Hitchin, and which in the case of $k=2$ is the Teichmüller space of the Riemann surface.

alg-geom↗

arXiv1994

Universal Hodge Bundle and Mumford Isomorphisms on the Abelian Inductive Limit of Teichmüller Spaces

Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain ``solenoidal Riemann surface'', $H_{\infty,abel}$. The main result constructs these universal $DET$ and Hodge line bundles on this direct limit, equips them with their Quillen metrics, and proves that they are related by the appropriate Mumford isomorphisms. Our work can be viewed as a contribution towards a non-perturbative formulation of the bosonic string theory, since it provides an interpretation of the Polyakov measure in a genus-independent fashion.

alg-geom↗