KK-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.Comment: Revised version to appear in Math. Nachrichte

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).Comment: 5 page

On the fundamental group of real toric varieties

Let $X(\Delta)$ be the real toric variety associated to a smooth fan $\Delta$. The main purpose of this article is: (i) to determine the fundamental group and the universal cover of $X(\Delta)$, (ii) to give necessary and sufficient conditions on $\Delta$ under which $\pi_1(X(\Delta))$ is abelian, (iii) to give necessary and sufficient conditions on $\Delta$ under which $X(\Delta)$ is aspherical, and when $\Delta$ is complete, (iv) to give necessary and sufficient conditions for \cc_{\Delta} to be a $K(\pi,1)$ space where \cc_{\Delta} is the complement of a real subspace arrangement associated to $\Delta$.Comment: 17 page

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.Comment: 41 pages, To appear in Transformation Group

Equivariant KK-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 $G/B$.Comment: Article revised based on referee's comments and journal reference adde

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.Comment: 12 page

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

Results on the topology of generalized real Bott manifolds

Generalized Bott manifolds (over $\mathbb C$ and $\mathbb R$) have been defined by Choi, Masuda and Suh. In this article we extend the results of arXiv:1609.05630 on the topology of real Bott manifolds to generalized real Bott manifolds. We give a presentation of the fundamental group, prove that it is solvable and give a characterization for it to be abelian. We further prove that these manifolds are aspherical only in the case of real Bott manifolds and compute the higher homotopy groups. Furthermore, using the presentation of the cohomology ring with $\mathbb Z_2$-coefficients, we derive a combinatorial characterization for orientablity and spin structure.Comment: 20 pages. This article derives heavily from arXiv:1609.05630 and uses the notations from arXiv:0803.274

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.Comment: 16 pages. Relation (ii)' in Defn 1.1 modified to correct an error in Lemma 2.2(i). Other minor errors correcte

On the existence of solutions to stochastic quasi-variational inequality and complementarity problems

Variational inequality problems allow for capturing an expansive class of problems, including convex optimization problems, convex Nash games and economic equilibrium problems, amongst others. Yet in most practical settings, such problems are complicated by uncertainty, motivating the examination of a stochastic generalization of the variational inequality problem and its extensions in which the components of the mapping contain expectations. When the associated sets are unbounded, ascertaining existence requires having access to analytical forms of the expectations. Naturally, in practical settings, such expressions are often difficult to derive, severely limiting the applicability of such an approach. Consequently, our goal lies in developing techniques that obviate the need for integration and our emphasis lies in developing tractable and verifiable sufficiency conditions for claiming existence. We begin by recapping almost-sure sufficiency conditions for stochastic variational inequality problems with single-valued maps provided in our prior work [44] and provide extensions to multi-valued mappings. Next, we extend these statements to quasi-variational regimes where maps can be either single or set-valued. Finally, we refine the obtained results to accommodate stochastic complementarity problems where the maps are either general or co-coercive. The applicability of our results is demonstrated on practically occurring instances of stochastic quasi-variational inequality problems and stochastic complementarity problems, arising as nonsmooth generalized Nash-Cournot games and power markets, respectively