Cohomological lifting of multi-toric graphs

TL;DR

This paper explores how to cohomologically lift multi-toric graphs associated with $G_{2}$-manifolds derived from circle bundles over symplectic $SU(3)$-manifolds, especially in the multi-Hamiltonian case, with applications in toric geometry.

Contribution

It introduces a cohomological method for lifting multi-moment graphs of $G_{2}$-structures from the base manifold, expanding understanding of their geometric and topological properties.

Findings

Cohomological lifting of multi-moment graphs is possible in the multi-Hamiltonian setting.

The procedure is explicitly illustrated within the framework of toric geometry.

The approach links $G_{2}$-geometry with combinatorial data from toric structures.

Abstract

We study $G_{2}$-manifolds obtained from circle bundles over symplectic $SU(3)$-manifolds with $T^{2}$-symmetry. When the geometry is multi-Hamiltonian, we show how the compact part of the resulting multi-moment graph for the $G_{2}$-structure may obtained cohomologically from the base. The lifting procedure is illustrated in the context of toric geometry.