Chern Characteristics and Todd-Hirzebruch Identities for Transpolar Pairs of Toric Spaces

Abstract

Standard toric geometry methods used to construct Calabi-Yau varieties may be extended to complete intersections in non-Fano varieties encoded by star triangulating non-convex polytopes. Similarly, mirror symmetry is conjectured to hold in terms of a transpolar duality generalizing the original construction of Batyrev and Borisov. The associated mirror pairs naturally include certain flip-folded, multi-layered multihedral objects, inclusively named VEX multitopes, and a correspondingly generalized transpolar duality. These self-overlaying VEX multitopes, long since known in pre-symplectic geometry, are found to correspond to certain non-algebraic but smooth toric spaces with Chern classes that satisfy the standard Todd-Hirzebruch identities. The computation of diffeomorphism invariants, including characteristic submanifold intersection numbers, corroborates their recent inclusion in the connected web of Calabi-Yau spaces and associated string compactifications: They arise together with the standard (Fano/reflexive polytope) constructions, within deformation families of generalized complete intersections in products of projective spaces.