Almost complex structures on connected sums of complex projective spaces

TL;DR

This paper proves that the connected sum of an odd number of complex projective spaces admits an almost complex structure, while even numbers do not, clarifying the topological constraints for such structures.

Contribution

It establishes a precise criterion for the existence of almost complex structures on connected sums of complex projective spaces, filling a gap in the understanding of their topology.

Findings

Connected sums with odd m admit almost complex structures.

Connected sums with even m do not admit such structures.

Provides a complete characterization based on the parity of m.

Abstract

We show that the m-fold connected sum $m\#\mathbb{C}\mathbb{P}^{2n}$ admits an almost complex structure if and only if m is odd.