Chern numbers of almost complex manifolds

Abstract

It is shown that any system of numbers that can be realised as the system of Chern numbers of an almost complex manifold of dimension 2n, n > 2, can also be realised in this way by a connected almost complex manifold. This answers an old question posed by Hirzebruch. Let -F(n) denote the number of partitions of the natural number n. A theorem of Milnor (cf. [5]) states that a system of ir(n) numbers can be realised as the system of Chern numbers of an almost complex manifold of (real) dimension 2rn if and only if it can be realised in this way by some algebraic manifold of (complex) dimension n belonging to some class A. (Manifolds are understood to be oriented, differentiable, and compact without boundary.) The class A is generated (under cartesian product and disjoint union) by the complex projective spaces, the hypersurfaces H(r,t) of double degree (1, 1) in Cpr x CPS with r, s > 1, and certain algebraic manifolds which realise the negative of the Chern numbers of the manifolds already listed. Thus, at least in principle, it is known which systems of ir(n) numbers can be realised as the Chern numbers of a 2n-dimensional almost complex manifold. In low dimensions, a complete set of restrictions is given as follows (cf. [5]): n=1: c1lOmod2, n = 2: c2+ C2-i0 mod 12, n = 3: C1C2=0 mod 24, c3lO-C3-0 mod 2, n =4: -c4l + 4cC2 +1CC3c+ 3C2 Cc4= _ mod720, 2c4 + cIc2 _ 0 mod 12, cic32c4 _ 0 mod 4. In [5] Hirzebruch raised the question whether a system of ir(n) numbers satisfying the necessary restrictions can be realised as the system of Chern numbers of a connected almost complex manifold of dimension 2n, and speculated that the connectedness assumption might impose additional inequalities between the Chern numbers. If the question is asked for complex or algebraic manifolds, there are indeed additional restrictions on the Chern numbers in the form of inequalities, as was first shown by Van de Ven [13] for complex dimension 2 (cf. [2]). In that paper Van de Ven also proved that no additional restrictions occur for connected almost complex manifolds of real dimension 4. Received by the editors May 2, 2000. 2000 Mathematics Subject Classification. Primary 57R20, 32Q60. (D2001 American Mathematical Society