155 research outputs found
Toric topology
We survey some results on toric topology.Comment: English translation of the Japanese article which appeared in
"Sugaku" vol. 62 (2010), 386-41
Torus graphs and simplicial posets
For several important classes of manifolds acted on by the torus, the
information about the action can be encoded combinatorially by a regular
n-valent graph with vector labels on its edges, which we refer to as the torus
graph. By analogy with the GKM-graphs, we introduce the notion of equivariant
cohomology of a torus graph, and show that it is isomorphic to the face ring of
the associated simplicial poset. This extends a series of previous results on
the equivariant cohomology of torus manifolds. As a primary combinatorial
application, we show that a simplicial poset is Cohen-Macaulay if its face ring
is Cohen-Macaulay. This completes the algebraic characterisation of
Cohen-Macaulay posets initiated by Stanley. We also study blow-ups of torus
graphs and manifolds from both the algebraic and the topological points of
view.Comment: 26 pages, LaTeX2e; examples added, some proofs expande
Toric Topology
Toric topology emerged in the end of the 1990s on the borders of equivariant
topology, algebraic and symplectic geometry, combinatorics and commutative
algebra. It has quickly grown up into a very active area with many
interdisciplinary links and applications, and continues to attract experts from
different fields.
The key players in toric topology are moment-angle manifolds, a family of
manifolds with torus actions defined in combinatorial terms. Their construction
links to combinatorial geometry and algebraic geometry of toric varieties via
the related notion of a quasitoric manifold. Discovery of remarkable geometric
structures on moment-angle manifolds led to seminal connections with the
classical and modern areas of symplectic, Lagrangian and non-Kaehler complex
geometry. A related categorical construction of moment-angle complexes and
their generalisations, polyhedral products, provides a universal framework for
many fundamental constructions of homotopical topology. The study of polyhedral
products is now evolving into a separate area of homotopy theory, with strong
links to other areas of toric topology. A new perspective on torus action has
also contributed to the development of classical areas of algebraic topology,
such as complex cobordism.
The book contains lots of open problems and is addressed to experts
interested in new ideas linking all the subjects involved, as well as to
graduate students and young researchers ready to enter into a beautiful new
area.Comment: Preliminary version. Contains 9 chapters, 5 appendices, bibliography,
index. 495 pages. Comments and suggestions are very welcom
The Geometry of T-Varieties
This is a survey of the language of polyhedral divisors describing
T-varieties. This language is explained in parallel to the well established
theory of toric varieties. In addition to basic constructions, subjects touched
on include singularities, separatedness and properness, divisors and
intersection theory, cohomology, Cox rings, polarizations, and equivariant
deformations, among others.Comment: 42 pages, 17 figures. v2: minor changes following the referee's
suggestion
Operational K-theory
We study the operational bivariant theory associated to the covariant theory
of Grothendieck groups of coherent sheaves, and prove that it has many
geometric properties analogous to those of operational Chow theory. This
operational K-theory agrees with Grothendieck groups of vector bundles on
smooth varieties, admits a natural map from the Grothendieck group of perfect
complexes on general varieties, satisfies descent for Chow envelopes, and is
A^1-homotopy invariant. Furthermore, we show that the operational K-theory of a
complete linear variety is dual to the Grothendieck group of coherent sheaves.
As an application, we show that the K-theory of perfect complexes on any
complete toric threefold surjects onto this group. Finally, we identify the
equivariant operational K-theory of an arbitrary toric variety with the ring of
integral piecewise exponential functions on the associated fan.Comment: 38 pages; v2: new exampes in Sections 5 and 7, and an new application
(Theorem 1.4), showing that the natural map from K-theory of perfect
complexes to the dual of the Grothendieck group of coherent sheaves is
surjective for complete toric threefolds; v3: final version published in
Documenta Mat
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