Connected sums of almost complex manifolds

TL;DR

This paper investigates conditions under which connected sums of almost complex manifolds admit almost complex structures, providing new results for sums involving complex projective spaces and their conjugates.

Contribution

It proves that certain connected sums of almost complex manifolds with complex projective spaces admit almost complex structures, and characterizes when sums of projective spaces and their conjugates do so.

Findings

Connected sums with (α-1) copies of CP^{2n} admit almost complex structures.

Adding a conjugate projective space to an almost complex manifold yields an almost complex structure.

The sum αCP^{2n} # βar{CP}^{2n} admits an almost complex structure if and only if α is odd.

Abstract

In this paper, firstly, for some $4n$-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 admits an almost complex structure, where $\alpha$ is a positive integer. Secondly, for a $2n$-dimensional almost complex manifold $M$, we get that $M\sharp \overline{\mathbb{C} P^{n}}$ also admits 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.