Classification of torus manifolds with codimension one extended actions

Abstract

The purpose of this paper is to classify torus manifolds (M 2n , T n ) with codimension one extended G-actions (M 2n , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T n . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.

Torus manifolds with non-abelian symmetries

“…We recover certain results of Kuroki [13,11,12] who gave a classification of torus manifolds with G-action and dim M/G ≤ 1 (see Corollaries 8.10 and 8.11).…”

“…Next we discuss torus manifolds M with G-action such that dim M/G ≤ 1. With Theorem 8.5, we recover the following two results of S. Kuroki [13,11]: Corollary 8.10. Let M be a simply connected torus manifold with G-action such that M is a homogeneous G-manifold.…”

Topological classification of torus manifolds which have codimension one extended actions

“…If 2k i1 + 2k i2 + 1 > 2ℓ (i = 1, 2, and ℓ = 2, 3), then these bundles are in the stable range, i.e., the dimension of fibre 2k i1 + 2k i2 + 1 is strictly greater than that of the base space CP ℓ ; therefore, it follows from the stable range theorem (see e.g. [8,Chapter 9]) that the two bundles are isomorphic (γ ⊗(−ρ 1 ) ) ⊕k 11…”

“…Let G be a compact, connected Lie group with maximal torus T n . In [9,10,11], the second author studied the torus manifolds which have extended G-actions. We call such a torus manifold a torus manifold with G-actions.…”

“…Let M be the set of simply connected torus manifolds M 2n with G-actions whose G-orbits have codimension one principal orbits. It follows from results in [10,11] that M consists of the following three types of manifolds:…”