On the chern numbers of certain complex and almost complex manifolds.

Let V be a compact, oriented 2n-dimensional (C--)differentiable manifold. If V admits a complex structure, compatible with its differentiable structure, the structural group of the tangent bundle of V can be restricted in a natural way from GL(2n,R) to GL(n,C). This gives for V a necessary condition of topological nature, which led Ch. Ehresmann and H. Hopf some 20 years ago to the concept of an almost complex structure, exactly meaning a restriction of the structural group of the tangent bundle of a differentiable manifold from GL(2n,R) to GL(n,C). An almost complex manifold is a differentiable manifold, provided with an almost complex structure. A complex manifold is of course an almost complex manifold, and for several years a condition has been known, the Eckmann-Frohlicher condition, which is necessary and sufficient for a C-y almost complex structure to be integrable, that is, to be induced by a complex structure. However, once there exists on a differentiable manifold an almost complex structure, by a slight change of this structure, an infinity of others can be obtained, and it would have been possible, though not likely, that to a given almost complex structure there is always a homotopic one, which is integrable. In particular, no example was known of a compact, differentiable manifold, admitting almost complex structures, but no complex structure whatsoever. Now to begin with, we would like to point out here that many examples of compact 4-dimensional differentiable manifolds, admitting almost complex structures, but no complex structure, can be obtained by combining a theorem of Ch. Ehresmann and W. W. Wu (in a formulation due to F. Hirzebruch and H. Hopf) with certain results of K. Kodaira. These last results-and consequently the main theorems of this announcement-depend in an essential and up to now unavoidable way on the Atiyah-Singer Riemann-Roch theorem. As will be explained at the end of this note, to obtain specific examples we need, apart from the theorems of Ehresmann-Wu and Atiyah-Singer, only some very general facts about complex surfaces. But we can get slightly more. To explain this properly, we shall recall the definition of the Chern numbers of a compact almost complex manifold V2'. Let [V] be the fundamental cycle of V, ci the ith Chern class of V, and il,. . . , in nonnegative integers with i1 + 2i2 + . . . + ni4 = n. Then clil U c2 J2 U ... U cnin C H2n (V,Z), and the value of this cohomology class on [V] is an integer. The 7r(n) integers, obtained this way for all i1, i2, .. . in with il + 2i2 + . . . + nin = n are the Chern numbers of V. J. Milnorl has determined the sets of 7r(n) integers occurring as the Chern numbers of a not necessarily connected V. It turns out in particular that they are the same as those occurring as the Chern numbers of the complex or even the projective algebraic manifolds of the same dimension, but again not necessarily connected. Milnor's methods do not allow us to decide which set