1,663 research outputs found
-theory of regular compactification bundles
Let be a connected reductive algebraic group. Let be a principal -bundle and be a regular
compactification of . We describe the Grothendieck ring of the associated
fibre bundle , as an algebra
over the Grothendieck ring of a canonical toric bundle over a flag bundle on
. These are relative versions of the results on equivariant
-theory of regular compactifications of . 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 -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 be the real toric variety associated to a smooth fan
. The main purpose of this article is: (i) to determine the fundamental
group and the universal cover of , (ii) to give necessary and
sufficient conditions on under which is abelian,
(iii) to give necessary and sufficient conditions on under which
is aspherical, and when is complete, (iv) to give
necessary and sufficient conditions for \cc_{\Delta} to be a space
where \cc_{\Delta} is the complement of a real subspace arrangement
associated to .Comment: 17 page
Equivariant K-theory of compactifications of algebraic groups
In this article we describe the -equivariant -ring of ,
where is a regular compactification of a connected complex reductive
algebraic group . Furthermore, in the case when is a semisimple group of
adjoint type, and its wonderful compactification, we describe its ordinary
-ring . More precisely, we prove that is a free module over
of rank the cardinality of the Weyl group. We further give an explicit
basis of over , and also determine the structure constants with
respect to this basis.Comment: 41 pages, To appear in Transformation Group
Equivariant -theory of flag varieties revisited and related results
In this article we obtain many results on the multiplicative structure
constants of -equivariant Grothendieck ring of the flag variety . We do
this by lifting the classes of the structure sheaves of Schubert varieties in
to , where denotes the representation
ring of the torus . We further apply our results to describe the
multiplicative structure constants of where is the
wonderful compactification of the adjoint group of , in terms of the
structure constants of Schubert varieties in the Grothendieck ring of .Comment: Article revised based on referee's comments and journal reference
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K-theory of quasi-toric manifolds
We describe the -ring of a quasi-toric manifold in terms of generators and
relations. We apply our results to describe the -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 and ) 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 -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
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