Publication

Chow Theory of Toric Variety Bundles

2026

Abstract

We describe the Chow homology and cohomology of toric variety bundles, with no restrictions on the singularities of the fibre. We present the ordinary and equivariant homologies as modules over the cohomology of the base, identify the ordinary cohomology with homology-valued Minkowski weights, and identify the equivariant cohomology with cohomology-weighted piecewise polynomial functions. We describe the product structure on Minkowski weights via a fan displacement rule, and the non-equivariant limit via equivariant multiplicities. Along the way we establish relative analogues of the K & uuml;nneth property and Kronecker duality. Applications include the balancing condition in logarithmic enumerative geometry.

Official source

https://infoscience.epfl.ch/entities/publication/6e392a8f-1ecf-477d-9811-3274f079132c

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2026

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/6e392a8f-1ecf-477d-9811-3274f079132c

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Related concepts (31)

Equivariant cohomology

In mathematics, equivariant cohomology (or Borel cohomology) is a cohomology theory from algebraic topology which applies to topological spaces with a group action. It can be viewed as a common generalization of group cohomology and an ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology ring with coefficient ring of the homotopy quotient : If is the trivial group, this is the ordinary cohomology ring of , whereas if is contractible, it reduces to the cohomology ring of the classifying space (that is, the group cohomology of when G is finite.

Cohomology

In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

Chow group

In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product.

Related publications (1)

Chow Theory of Toric Variety Bundles

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Une dégustation topologique: homotopy theory in the Swiss Alps : Proceedings of the Arolla conference on algebraic topology, Arolla, Switzerland, August 25-September 1, 1999.