Toric Constructions via Integral Transversality
Abstract
In the Delzant world, symplectic toric manifolds correspond to unimodular polytopes, and unimodularity-preserving polytope constructions correspond to geometric constructions on the manifold side. We treat, in a unified framework, constructions such as taking a face, taking a slice, and polytope intersections, identifying each with its geometric counterpart, namely invariant submanifolds, symplectic reduction, and reduction of a product (a full list appears in Table 1). The key tool is "integral transversality", a lattice-refined transversality condition that provides a uniform criterion for preserving unimodularity, ensuring smoothness on the manifold side. To keep every construction within a single ambient vector space -- the dual of the Lie algebra of a fixed torus -- we allow the torus actions involved to have nontrivial, but connected kernels.
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BibTeX
@misc{silva2026toric,
title = {{Toric Constructions via Integral Transversality}},
author = {Ana Cannas da Silva and Reto Kaufmann},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31026},
eprint = {2608.31026},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31026},
}
RIS
TY - UNPB
TI - Toric Constructions via Integral Transversality
AU - Ana Cannas da Silva
AU - Reto Kaufmann
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31026
UR - https://arxiv.org/abs/2608.31026
AB - In the Delzant world, symplectic toric manifolds correspond to unimodular polytopes, and unimodularity-preserving polytope constructions correspond to geometric constructions on the manifold side. We treat, in a unified framework, constructions such as taking a face, taking a slice, and polytope intersections, identifying each with its geometric counterpart, namely invariant submanifolds, symplectic reduction, and reduction of a product (a full list appears in Table 1). The key tool is "integral transversality", a lattice-refined transversality condition that provides a uniform criterion for preserving unimodularity, ensuring smoothness on the manifold side. To keep every construction within a single ambient vector space -- the dual of the Lie algebra of a fixed torus -- we allow the torus actions involved to have nontrivial, but connected kernels.
ER -