Almost complex structures which are compatible with Kähler or symplectic structures (Q1370906)

This paper contains two results related to the following questions. Given an almost complex structure \(J\) on a compact oriented 4-manifold \(M\), does \(M\) admit a compatible symplectic structure? If \(J\) is integrable, does \(M\) admit a compatible Kähler structure? In fact, thinking of \(J\) as a section of a principal \(SO(4)/U(2)\) bundle over \(M\), these questions can be posed for the homotopy class represented by \([J]\). The results of this paper show that there are always homotopy classes of almost complex structures for which the answer to these questions is negative. The results make use of a free involution \(p\), defined by \textit{S. Donaldson} [J. Differ. Geom. 26, 397-428 (1987; Zbl 0683.57005)] on the set of homotopy classes of almost complex structures. The precise statements are as follows. Theorem: Let \(M\) be a closed oriented 4-manifold such that \(b_+^2(M)\geq 2\), or \(b_+^2(M)=1\) and \(b_1(M)=0\), or \(M\) is diffeomorphic to a ruled or hyperelliptic surface. Suppose that a homotopy class \([J]\) is compatible with a symplectic structure. Then the homotopy class \(p[J]\) is not compatible with any symplectic structure. Theorem: Let \(M\) be a closed oriented 4-manifold. Suppose that a homotopy class \([J]\) is compatible with a Kähler structure. Then the homotopy class \(p[J]\) is not compatible with any Kähler structure. The proofs are via an application of C. H. Taubes' results (refined by D. Salamon to remove a sign ambiguity) on Seiberg-Witten invariants for symplectic 4-manifolds.