arXiv subjects

V. Uma

Publications and source records attributed to V. Uma.

18 recordsLinked to original sources

arXiv2025

Equivariant $K$-theory of cellular toroidal embeddings

In this article we describe the $G_{comp}\times G_{comp}$-equivariant topological $K$-ring of a {\em cellular} toroidal embedding $\mathbb{X}$ of a complex connected reductive algebraic group $G$. In particular, our results extend the results in \cite{u1} and \cite{u2} on the regular embeddings of $G$, to the equivariant topological $K$-ring of a larger class of (possibly singular) cellular toroidal embeddings. They are also a topological analogue of the results in \cite{gon} on the operational equivariant algebraic $K$-ring, for cellular toroidal embeddings.

math.AG↗

arXiv2024

Equivariant $K$-theory of cellular toric bundles and related spaces

In this article we describe the equivariant and ordinary topological $K$-ring of a toric bundle with fiber a $T$-{\it cellular} toric variety. This generalizes the results in \cite{su} on $K$-theory of smooth projective toric bundles. We apply our results to describe the equivariant topological $K$-ring of a toroidal horospherical embedding.

math.KT↗

arXiv2024

Equivariant $K$-theory of cellular toric varieties

In this article we describe the $T_{comp}$-equivariant topological $K$-ring of a $T$-{\it cellular} complete toric variety. We further show that $K_{T_{comp}}^0(X)$ is isomorphic as an $R(T_{comp})$-algebra to the ring of piecewise Laurent polynomial functions on the associated fan denoted $PLP(\Delta)$. Furthermore, we compute a basis for $K_{T_{comp}}^0(X)$ as a $R(T_{comp})$-module and multiplicative structure constants with respect to this basis.

math.KT↗

arXiv2022

Equivariant Grothendieck ring of a complete symmetric variety of minimal rank

We describe the $G$-equivariant Grothendieck ring of a regular compactification $X$ of an adjoint symmetric space $G/H$ of minimal rank. This extends the results of Brion and Joshua for the equivariant Chow ring of wonderful symmetric varieties of minimal rank and generalizes the results by the author on the regular compactification of an adjoint semisimple group.

math.AG↗

arXiv2021

Equivariant K-theory of toric orbifolds

Toric orbifolds are a topological generalization of projective toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure the existence of an invariant cell structure on it and call such a toric orbifold retractable. In this paper, our main goal is to study equivariant cohomology theories of retractable toric orbifolds. Our results extend the corresponding results on divisive weighted projective spaces.

math.AG↗

arXiv2020

$K$-theory of regular compactification bundles

Let $G$ be a connected reductive algebraic group. Let $\mathcal{E}\rightarrow \mathcal{B}$ be a principal $G\times G$-bundle and $X$ be a regular compactification of $G$. We describe the Grothendieck ring of the associated fibre bundle $\mathcal{E}(X):=\mathcal{E}\times_{G\times G} X$, as an algebra over the Grothendieck ring of a canonical toric bundle over a flag bundle on $\mathcal{B}$. These are relative versions of the results on equivariant $K$-theory of regular compactifications of $G$. They also generalize the well known results on the Grothendieck rings of projective bundles, toric bundles and flag bundles.

math.AG↗

arXiv2020

Article citation study: Context enhanced citation sentiment detection

Citation sentimet analysis is one of the little studied tasks for scientometric analysis. For citation analysis, we developed eight datasets comprising citation sentences, which are manually annotated by us into three sentiment polarities viz. positive, negative, and neutral. Among eight datasets, three were developed by considering the whole context of citations. Furthermore, we proposed an ensembled feature engineering method comprising word embeddings obtained for texts, parts-of-speech tags, and dependency relationships together. Ensembled features were considered as input to deep learning based approaches for citation sentiment classification, which is in turn compared with Bag-of-Words approach. Experimental results demonstrate that deep learning is useful for higher number of samples, whereas support vector machine is the winner for smaller number of samples. Moreover, context-based samples are proved to be more effective than context-less samples for citation sentiment analysis.

cs.CL↗

arXiv2018

Cohomology of torus manifold bundles

Let $X$ be a torus manifold with locally standard action of a compact torus $T$ of half the dimension and orbit space a homology polytope. Smooth complete complex toric varieties and quasi-toric manifolds are examples of torus manifolds. Consider a principal bundle with total space $E$ and base $B$ with fibre and structure group $T$. Let $E(X)$ denote the total space of the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as an algebra over the singular cohomology ring of $B$ and a presentation of the topological $K$-ring of $E(X)$ as an algebra over the topological $K$-ring of $B$. These are relative versions of the results of M. Masuda and T. Panov [13] on the cohomology ring of a torus manifold and P. Sankaran [14] on the topological $K$-ring of a torus manifold. Further, they extend the results due to P. Sankaran and V. Uma [15] on the cohomology ring and topological $K$-ring of toric bundles with fibre a smooth projective toric variety, to a toric bundle with fibre any smooth complete toric variety.

math.KT↗

arXiv2018

$K$-theory of $\mbox{hyperKähler}$ toric manifolds

Let $X$ be a toric $\mbox{hyperKähler}$ manifold. The purpose of this note is to describe the topological $K$-ring $K^*(X)$ of $X$. We give a presentation for the topological $K$-ring in terms of generators and relations similar to the known description of the cohomology ring of these manifolds.

math.AT↗

arXiv2018

Equivariant $K$-theory of quasitoric manifolds

Let $X(Q,Λ)$ be a quasitoric manifold associated to a simple convex polytope $Q$ and characteristic function $Λ$. Let $T\cong (\mathbb{S}^1)^n$ denote the compact $n$-torus acting on $X=X(Q,Λ)$. The main aim of this article is to give a presentation of the $T$-equivariant $K$-ring of $X$, as a Stanley-Reisner ring over $K^*(pt)$. We also derive the presentation for the ordinary $K$-ring of $X$.

math.AT↗

arXiv2016

Some results on the topology of real Bott towers

The main aim of this article is to study the topology of real Bott towers as special and interesting examples of real toric varieties. We first give a presentation of the fundamental group of a real Bott tower and show that the fundamental group is abelian if and only if the real Bott tower is a product of circles. We further prove that the fundamental group of a real Bott tower is always solvable and it is nilpotent if and only if it is abelian. We then describe the cohomology ring of a real Bott tower and also give recursive formulae for the Steifel Whitney classes. We derive combinatorial characterization for orientability of these manifolds and further give a combinatorial formula for the $(n-1)$th Steifel Whitney class. In particular, we show that if a Bott tower is orientable then the $(n-1)$th Steifel Whitney class must also vanish. Moreover, by deriving a combinatorial formula for the second Steifel-Whitney class we give a necessary and sufficient condition for the Bott tower to admit a spin structure. We finally prove the vanishing of all the Steifel-Whitney numbers and hence establish that these manifolds are null-cobordant.

math.AT↗

arXiv2014

Equivariant $K$-theory of flag varieties revisited and related results

In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.

math.AG↗

arXiv2014

Equivariant $K$-theory of regular compactifications: further developments

In this article we describe the $\tG\times \tG$-equivariant $K$-ring of $X$, where $\tG$ is a {\it factorial} cover of a connected complex reductive algebraic group $G$, and $X$ is a regular compactification of $G$. Furthermore, using the description of $K_{\tG\times \tG}(X)$, we describe the ordinary $K$-ring $K(X)$ as a free module of rank the cardinality of the Weyl group, over the $K$-ring of a toric bundle over $G/B$, with fibre the toric variety $\bar{T}^{+}$, associated to a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactification (see \cite{u}). Further, we give an explicit presentation of $K_{\tG\times \tG}(X)$ as well as $K(X)$ as an algebra over the $K_{\tG\times \tG}(\bar{G_{ad}})$ and $K(\bar{G_{ad}})$ respectively, where $\bar{G_{ad}}$ is the wonderful compactification of the adjoint semisimple group $G_{ad}$. Finally, we identify the equivariant and ordinary Grothendieck ring of $X$ respectively with the corresponding rings of a canonical toric bundle over $\bar{G_{ad}}$ with fiber the toric variety $\bar{T}^+$.

math.AG↗

arXiv2010

Cobordism ring of toric varieties

We describe the equivariant cobordism ring of smooth toric varieties. This equivariant description is used to compute the ordinary cobordism ring of such varieties.

math.AG↗

arXiv2007

Equivariant K-theory of compactifications of algebraic groups

In this article we describe the $G\times G$-equivariant $K$-ring of $X$, where $X$ is a regular compactification of a connected complex reductive algebraic group $G$. Furthermore, in the case when $G$ is a semisimple group of adjoint type, and $X$ its wonderful compactification, we describe its ordinary $K$-ring $K(X)$. More precisely, we prove that $K(X)$ is a free module over $K(G/B)$ of rank the cardinality of the Weyl group. We further give an explicit basis of $K(X)$ over $K(G/B)$, and also determine the structure constants with respect to this basis.

math.AG↗

arXiv2006

K-theory of torus manifolds

The {\it torus manifolds} have been defined and studied by M. Masuda and T. Panov (arXiv:math.AT/0306100) who in particular describe its cohomology ring structure. In this note we shall describe the topological $K$-ring of a class of torus manifolds (those for which the orbit space under the action of the compact torus is a {\it homology polytope} whose {\it nerve} is a {shellable} simplicial complex) in terms of generators and relations. Since these torus manifolds include the class of quasi-toric manifolds this is a generalisation of earlier results due to the author and P. Sankaran (arXiv: math.AG/0504107).

math.AT↗

arXiv2005

Cohomology of toric bundles

We describe the singular cohomology ring, the K-ring of complex vector bundles, the Chow ring, and the Grothendieck ring of coherent sheaves of the total space of the fibre bundle with base space an irreducible nonsingular complete Noetherian scheme and fibre a nonsingular projective T-toric variety associated to a prinicipal T-bundle over the field of complex numbers.

math.AG↗

arXiv2005

K-theory of quasi-toric manifolds

We describe the $K$-ring of a quasi-toric manifold in terms of generators and relations. We apply our results to describe the $K$-ring of Bott-Samelson varieties.

math.AG↗