A class of torus manifolds with nonconvex orbit space

TL;DR

This paper explores a new class of smooth torus manifolds with orbit spaces resembling polytopes with holes, constructing associated moment angle manifolds, and analyzing their complex structures and topological invariants.

Contribution

It introduces a construction for moment angle manifolds for polytopes with holes and establishes conditions for stable almost complex structures on the associated torus manifolds.

Findings

Existence of stable almost complex structures on these manifolds

A combinatorial formula for the Hirzebruch χ_y genus

Examples of almost complex manifolds without complex structures

Abstract

We study a class of smooth torus manifolds whose orbit space has the combinatorial structure of a simple polytope with holes. We construct moment angle manifolds for such polytopes with holes and use them to prove that the associated torus manifolds admit stable almost complex structure. We give a combinatorial formula for the Hirzebruch $\chi_y$ genus of these torus manifolds. We show that they have (invariant) almost complex structure if they admit positive omniorientation. We give examples of almost complex manifolds that do not admit a complex structure. When the dimension is four, we calculate the homology groups and describe a method for computing the cohomology ring.