Almost complex structure and the quotient four-manifold by an anti-symplectic involution
The authors study the quotient of an oriented, closed, symplectic four-manifold \(X\) under an anti-symplectic involution \(\sigma\) with two-dimensional compact fixed point submanifold \(X^\sigma\) (not necessarily orientable or connected). They show that if \(b_2^+(X) \not \equiv b_1(X) + 3 \mod 4,\) then the quotient \(X/\sigma\) does not admit any almost complex structure. They prove a converse holds if we restrict to simply connected \(X\) or \(X\) Kähler and \(\sigma\) anti-holomorphic. Important tools in the authors' arguments are the fundamental classification theorems on quadratic forms of smooth 4-manifolds due to \textit{V. A. Rokhlin} [Dokl. Akad. Nauk SSSR, n. Ser. 84, 221--224 (1952; Zbl 0046.40702)], \textit{S. K. Donaldson} [J. Differ. Geom. 18, 279-315 (1983; Zbl 0507.57010)] and \textit{J.-H. Kim} [Math. Ann. 329, No.~1, 31--47 (2004; Zbl 1070.57021)] as well as the criteria of \textit{F. Hirzebruch} and \textit{H. Hopf} [Math. Ann. 136, 1566--172 (1958; Zbl 0088.39403)] for the existence of an almost complex structure. An important motivation for the work in the paper is an open conjecture of Akbulut that the quotient is completely decomposable, i.e. \(X/\sigma \simeq \#r \mathbb C \mathbb P^2 \# s\overline{\mathbb C \mathbb P}^2\) or \(\# n (S^2 \times S^2)\) for some \(r,s,n\). \textit{S. Akbulut} [J. Reine Angew. Math. 447, 83--90 (1994; Zbl 0861.57027)] proved this with the additional hypotheses that \(X\) is a complex algebraic surface and \(\sigma\) is the complex conjugation with a real algebraic surface as a fixed point set.
scientific article; zbMATH DE number 5647959