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

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## 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.

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