Almost complex structures on connected sums of almost complex manifolds and complex projective spaces

Abstract

In this paper, the existence of almost complex structures on connected sums of almost complex manifolds and complex projective spaces are investigated. Firstly, we show that if M is a 2n-dimensional almost complex manifold, then so is $$M\sharp \overline{{\mathbb {C}}P^{n}}$$ , where $$\overline{{\mathbb {C}}P^{n}}$$ is the n-dimensional complex projective space with the reversed orientation. Secondly, for any positive integer $$\alpha $$ and any 4n-dimensional almost complex manifolds $$M_{i}, ~1\le i \le \alpha $$ , we prove that $$\left( \sharp _{i=1}^{\alpha } M_{i}\right) \sharp (\alpha {-}1) {\mathbb {C}}P^{2n}$$ must admit an almost complex structure. At last, as an application, we obtain that $$\alpha {\mathbb {C}}P^{2n}\sharp ~\beta \overline{{\mathbb {C}}P^{2n}}$$ admits an almost complex structure if and only if $$\alpha $$ is odd.