Topology versus Chern Numbers for Complex 3-folds

of the almost-complex manifold (X, J). The only obstruction [10] to the existence of an almost-complex structure J on X is that X be spinc. This happens precisely when the second Stiefel-Whitney class w2(X) ∈ H(X,Z2) can be written as the mod-2 reduction of an element of H2(X,Z), in which case each preimage of w2 in H2(X,Z) can be realized as c1 for some almostcomplex structure J . It follows that the Chern numbers c1 and c1c2 of the almost-complex (X, J) are certainly not topological invariants of the 6-manifold X. For example, if X = CP3, every integer of the form 8j can be realized as c1c2, and every integer of the form 8j3 can be realized as c1 for some almost-complex structure J on CP3. On the other hand, c3 is the Euler class of TX, so that c3 = χ(X) is actually a homotopy invariant of X.