Complex and almost-complex structures on six dimensional manifolds

Abstract

We investigate the properties of hypothetical exotic complex structures on three dimensional complex projective space CP³. This is motivated by the long standing question in differential geometry of whether or not the six sphere S⁶ admits an integrable almost-complex structure. An affirmative answer to this question would imply the existence of many exotic complex structures on CP³. It is known that CP³ admits many topologically different almost-complex structures, but it is unknown whether or not CP³ admits an integrable almost-complex structure other that the standard Kaḧler structure. In this manuscript we give lower bounds on the Hodge numbers of hypothetical exotic structures on CP³ and a necessary condition for the Frol̈icher spectral sequence to degenerate at the second level. We also give topological constraints on the classes of hypothetical exotic complex structures which areC*-symmetric. We give restrictions on the fixed point sets of such C* actions.