Chern numbers of the Chern submanifolds

We solve the generalized Milnor–Hirzebruch problem on the relations between Chern numbers of an arbitrary stably almost complex manifold and Chern numbers of its virtual Chern submanifolds. As a corollary we derive new divisibility conditions for Chern classes of an arbitrary complex vector bundle over a stably almost complex manifold. Introduction The generalized Milnor–Hirzebruch problem was posed in [5] where the authors found a non-trivial relation between signatures of an arbitrary almost complex manifold and its virtual submanifolds dual to the tangent Pontrjagin classes in complex cobordism. The general question is: What are all the relations between an arbitrary stably almost complex manifold and its virtual Chern submanifolds (note that all characteristic classes of any stably almost complex manifold can be expressed in terms of Chern classes)? This question is a part of a most general question on the relations between manifold and any element of its cobordism ring. Recall that any cobordism class of a smooth manifold M can be realized as a smooth submanifold L ⊂ M × R with corresponding structure in the normal bundle of embedding for N large enough. Thus, any cobordism class defines up to bordism a smooth submanifold in M ×R . We call such submanifold as a virtual submanifold of M . In many cases virtual submanifolds arise from canonical submanifolds of the origin. For example, virtual submanifold corresponding to Euler class of an arbitrary vector bundle can be realized by the zero set of any generic section. It is natural that virtual submanifolds inherit some properties from the original manifold. In the present paper we establish the general relations between Chern numbers of an arbitrary stably almost complex manifold and its virtual submanifolds defined by Chern classes in complex cobordisms of an arbitrary complex vector bundle. The result of [5] becomes now a particular case of our general theorem. ∗Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh, Scotland, EH9 3JZ; e-mail: [email protected]