Linear SU(n)-actions on complex projective spaces

Let U* be the bordism ring of weakly complex manifolds and let G be a compact Lie group. Denote by SF(G), an ideal in U* of those bordism classes represented by a weakly complex manifold on which the group G acts smoothly without stationary points and the action preserves a weakly complex structure. For a compact abelian Lie group G the ideal SF(G) was computed by torn Dieck [8]. Such ideals are similarly defined in the bordism ring Ω* of oriented manifolds and those were computed for certain abelian groups by Floyd [3] and Stong [7]. But it seems that there is no useful method to compute the ideal SF(G) for a non-abelian Lie group G. First we give an upper bound and a lower bound of SF(G) for any compact Lie group G. To state our result precisely we introduce some notations as follows. Denote by I(G), a. set of positive integers such that n^I(G) if and only if there is an w-dimensional complex G-vector space without G-invariant onedimensional subspaces, by m(G) the maximum dimension of proper closed subgroups of G, and put