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 ℂPn but
not biholomorphic to ℂPn. 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.
