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

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 . artial converse if X is simply-connected and b 2 + ( X ) ≡ 3 mod 4 , then the X / σ admits an almost complex structure. e 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 .