19 Sep 2002

Boris Kruglikov "Existence and non-existence results in almost complex geometry - II".

In this lecture I will show there are no higher-dimensional analogues for submanifolds in almost complex category. Namely a generic almost complex manifold (requirements being explicitly specified) will be shown to possess neither of the following: proper PH-submanifolds except PH-curves, PH-automorphisms different from the identity, non-trivial PH-submersions. To my knowledge though the result was expected, no proof has appeared anywhere before.
Thus the Gromov's theory  of pseudoholomorphic curves is the only existing almost complex analogue of the complex theory in exact sense. So to get other tools one needs an idea of approximate PH-map introduced by Donaldson, that we will describe at the end of the talk.