Weakly complex manifolds with semi-free \(S^ 1\)-action whose fixed point set has complex codimension 2 (Q1190715)

Let \(M\) be a closed weakly complex manifold of dimension \(2n\) and let \(c_ 1\) denote the first Chern class of the weakly complex structure of \(M\). The author proves the following theorem: If \(M\) admits a smooth semi-free action of the circle group \(S^ 1\), which preserves the complex structure, and if all components of the fixed point set have the same complex codimension \(2k\), then \(c^ n_ 1[M]\equiv 0\bmod (2k)^{2k}\). This result is used to deduce conditions for \(M\) which imply the existence of a closed weakly complex manifold \(N\) of dimension \(2n\) with the following properties: \(N\) admits a smooth semi-free \(S^ 1\)-action preserving the complex structure such that all components of the fixed point set have complex codimension 2 and \([M]=[N]\) in \({\mathcal U}_ *\otimes \mathbb{Z}_ 2\), the complex bordism ring \({}\bmod 2\).