\(K\)-rings of smooth complete toric varieties and related spaces (Q1004655)

In earlier work with \textit{V. Uma} [Osaka J. Math. 44, No.~1, 71--89 (2007; Zbl 1125.55003)], the author gave a description of the \(K\)-ring of a quasi-toric manifold in terms of generators and relations. The purpose of this paper is to give a similar description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope (Theorem 5.3). Using a method presented in the above paper, \textit{V. Uma} has previously shown in [Contemporary Mathematics 460, 385--389 (2008; Zbl 1151.55005)] that the same result as that obtained here holds under the additional assumption that these torus manifolds satisfy a shellability condition. The author presents here an approach to avoid this shellability condition. The paper consists of two parts, the first of which gives a description of the \(K\)-ring of a non-singular complete toric variety in terms of generators and relations (Theorem 2.2). In the second part, it is shown that one can obtain the same result for a torus manifold by carrying out exactly all the steps of the proof corresponding to the above case. Then one uses results of \textit{M. Masuda} and \textit{T. Panov} [Osaka J. Math. 43, No.~3, 711--746 (2006; Zbl 1111.57019)]. But in fact, the second part subsumes the first, since the class of torus manifolds includes complete nonsingular toric varieties as well as quasi-toric manifolds. The result obtained can be summarized as follows. Let \(X\) be a smooth compact oriented connected manifold with an effective smooth action of a torus \(T\) such that \(X^T\neq \phi\). From a local standardness condition given one finds that there are only finitely many codimension 2 submanifolds \(V_1, \dots, V_d\), each of which is pointwise fixed by a circle subgroup of \(T\). These circle subgroups detemine \(v_1, \dots, v_d \in \text{Hom}({\mathbb{S}}^1, T)=H_2(BT)\). Let \(Q_i\) denote the image of \(V_i\) under the quotient map from \(X\) to \(Q=X/T\). (This orbit space becomes a homology polytope with \(d\) facets \(Q_1, \dots, Q_d\).) Then it follows that \(K(X)\) is isomorphic to the ring \(\mathbb{Z}[y_1, \dots, y_d]/{\mathcal I}\), where \({\mathcal I}\) is the ideal generated by the elements \(y_{j_1}, \dots, y_{j_k}\), whenever \(\bigcap_{i=1}^kQ_j=\phi\) in \(Q\), and \[ \prod_{j, \langle u, v_j \rangle >0}(1-y_j)^{\langle u, v_j \rangle} -\prod_{j, \langle u, v_j \rangle <0}(1-y_j)^{-\langle u, v_j \rangle} \] for \(u \in H^2(BT)\).