Quaternionic toric varieties
Let \(H\) be the skew field of (Hamilton's) quaternions, and \(G\) the multiplicative group of quaternions of norm 1 so that \(G\) coincides with the 3-sphere \(S^3\). Among more general ``quaternionic spaces'', the author defines ``quaternionic toric varieties'', which are neither algebraic varieties nor spaces with actions of products of copies of \(G\), but which nevertheless share some of the features of complex (resp. real) toric projective varieties arising out of convex polytopes, where the complex unit circle \(S^1\) (resp. \(S^0 = \{\pm 1\})\) is supposed to play the role of \(G\). More specifically, let \(P\) be an \(n\)-dimensional convex polytope. An \(n\)-dimensional quaternionic toric variety \(X\) is a real \(4n\)-dimensional compact Hausdorff space obtained as a quotient of \(P \times G^n\) with respect to an equivalence relation specified by a ``characteristic function'' \(\lambda\). Roughly speaking, \(\lambda\) assigns for each face \(F\) of the polytope \(P\) a map \(\lambda (F) : G^n \leftarrow G^{\dim F}\) obtained as the composite, with the projection to the first \(\dim F\) factors, of a self-diffeomorphism \(\varphi : G^n \leftarrow G^n\) such that both \(\varphi\) and \(\varphi^{- 1}\) are expressed in terms of \(n\)-tuples of words in the \(n\) coordinates of \(G^n\). If \(F'\) is a face of \(F\), then \(\lambda (F')\) and \(\lambda (F)\) are supposed to satisfy suitable compatibility conditions. Then \((p, s_1, \dots, s_n)\) and \((p, t_1, \dots, t_n)\) in \(F \times G^n\) are defined to be equivalent if \(\lambda (F) (s_1, \dots, s_n) = \lambda (F) (t_1, \dots, t_n)\). The \(n\)-dimensional quaternionic projective space \(HP^n\) is a quaternionic toric variety when \(P\) is the \(n\)-dimensional simplex with a suitable \(\lambda\). \(SO(3)\) acts on \(G\) through inner automorphisms of quaternions, hence acts continuously on quaternionic toric varieties. The author determines when a quaternionic toric variety is nonsingular (i.e., a topological manifold), and determines its cohomology ring as in the theorem of Jurkiewicz and Danilov in the case of complex toric varieties. These nonsingular quaternionic toric varieties are analogous to the manifolds considered by \textit{M. W. Davis} and \textit{T. Januszkiewicz} [Duke Math. J. 62, No. 2, 417-451 (1991; Zbl 0733.52006)]. The author then goes on to study the case \(n = 2\), and obtains interesting results on topological 8-manifolds involving smoothability, exotic 7-spheres and \(SO (3)\)-actions.
scientific article; zbMATH DE number 815767
scientific article