Chow theory of toric variety bundles
Francesca Carocci, Leonid Monin, Navid Nabijou
TL;DR
This work develops the Chow theory of toric variety bundles Y → X with arbitrary fibre singularities by extending absolute toric results to a relative setting. It provides a complete framework: A^*_T Y is captured by cohomology-weighted piecewise polynomials PP^*(Σ) ⊗ A^*X, while A^*Y is described as a module of Minkowski weights MW^*(Σ, L, X) with a precise product via a fan displacement rule. Key contributions include relative forms of the Künneth property and Kronecker duality, a canonical product rule for Minkowski weights, and a systematic bridge from equivariant piecewise polynomials to non-equivariant Minkowski weights using equivariant multiplicities. Applications to logarithmic enumerative geometry and toroidal horospherical varieties demonstrate the practical impact of the theory and its potential to inform the balancing conditions and degenerations in complex geometric settings.
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 Künneth property and Kronecker duality. Applications include the balancing condition in logarithmic enumerative geometry.