Actions of special unitary groups on a product of complex projective spaces
Let X be a connected closed orientable \(C^{\infty}\) manifold which admits a nontrivial smooth SU(n) action. Suppose \[ H^*(X,{\mathbb{Q}})={\mathbb{Q}}[u,v]/(u^{a+1},v^{b+1}),\quad \deg u=\deg v=2, \] that is, the cohomology ring of X is isomorphic to that of a product \(P_ a(C)\times P_ b(C)\) of complex projective spaces, where \({\mathbb{Q}}\) is the field of rational numbers. For such situation the author shows: Theorem. Suppose \(1\leq b\leq a<n\leq a+b\leq 2n-3\). Then, \(a=n-1\) and X is equivariantly diffeomorphic to \(P_{n-1}(C)\times Y\), where Y is a connected closed orientable manifold whose rational cohomology ring is isomorphic to that of \(P_ b(C)\), and SU(n) acts naturally on \(P_{n- 1}(C)\) and trivially on Y.
scientific article; zbMATH DE number 3846490