The foundations of $(2n,k)$-manifolds

TL;DR

This paper develops a foundational axiomatic framework for $(2n,k)$-manifolds, generalizing toric geometry, and demonstrates its application through the example of complex Grassmann manifolds with torus actions.

Contribution

It introduces a system of axioms for $(2n,k)$-manifolds, constructs a model space with torus action, and extends the theory beyond toric cases to homogeneous spaces with non-zero complexity.

Findings

Established a model space $rak{E}$ with a $T^k$-action for $(2n,k)$-manifolds.

Proved the existence of a $T^k$-equivariant homeomorphism between $rak{E}$ and $M^{2n}$.

Applied the theory to complex Grassmann manifolds $G_{k+1,q}$ with torus actions.

Abstract

In the focus of our paper is a system of axioms that serves as a basis for introducing structural data for $(2n,k)$-manifolds $M^{2n}$, where $M^{2n}$ is a smooth, compact $2n$-dimensional manifold with a smooth effective action of the $k$-dimensional torus $T^k$. In terms of these data a construction of the model space $\mathfrak{E}$ with an action of the torus $T^k$ is given, such that there exists a $T^k$-equivariant homeomorphism $\mathfrak{E}\to M^{2n}$. This homeomorphism induces a homeomorphism $\mathfrak{E}/T^k\to M^{2n}/T^k$. The number $d=n-k$ is called the complexity of an $(2n,k)$-manifold. Our theory comprises toric geometry and toric topology, where $d=0$. It is shown that the class of homogeneous spaces $G/H$ of compact Lie groups, where rk$G=$rk$H$, contains $(2n,k)$-manifolds that have non zero complexity. The results are demonstrated on the complex Grassmann manifolds $G_{k+1,q}$ with an effective action of the torus $T^k$.