Topological invariants of putative ℂ∗-symmetric exotic complex projective spaces

Abstract

A classical problem in complex geometry is to determine the conditions under which two manifolds with the same differentiable structure admit different complex structures. We call a complex manifold X an exotic complex projective space if it is diffeomorphic to ℂP n but not biholomorphic to ℂP n . It is unknown whether such exotic structures exist, but Emery Thomas has given necessary and sufficient conditions for an element of the cohomology ring to occur as the total Chern class of an almost-complex structure in low dimensions, thus establishing the existence of almost-complex structures with exotic Chern classes. We show that most of these elements cannot occur as the total Chern class of a complex structure with ℂ* symmetry. We include an overview of the equivariant index theory used in the proof.