Remarks on the Complex Structures on ${\mathbb{P}}^3$ and $S^2\times S^4$

Abstract

Assuming the validity of the recently proposed \textit{``A compact complex threefold fibred by tori over the projective line, and the six-sphere''}, we construct an exotic complex structure on ${\mathbb{P}}^3$, distinct from the point-blowup structures of Huckleberry, Kebekus, and Peternell. Then we perform an Atiyah flop to produce a complex structure on the standard smooth manifold $S^2\times S^4$.