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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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