Almost complex structure and the quotient four-manifold by an anti-symplectic involution

Abstract

Suppose that X is a closed, symplectic four-manifold with an anti-symplectic involution σ and its two-dimensional fixed point set. We show that the quotient X / σ admits no almost complex structure if b 2 + (X) ≢ b 1 (X) + 3 mod 4. As a partial converse if X is simply-connected and b 2 + (X) ≡ 3 mod 4, then the X / σ admits an almost complex structure. Also we show that the quotient X / σ admits an almost complex structure if X is Kähler and b 2 + (X) ≡ b 1 (X) + 3 mod 4. © 2009 Elsevier B.V. All rights reserved.