Non-extendability of complex structures

TL;DR

This paper demonstrates that certain local complex structures on a subset of the 6-sphere can be extended globally as almost complex structures, but these extensions cannot be integrable, highlighting limitations in deformation strategies for complex structures on spheres.

Contribution

It proves the non-extendability of integrable complex structures on the 6-sphere, showing that local structures cannot be globally deformed into integrable ones.

Findings

Existence of a complex structure on a subset of S^6

Extension of this structure to a global almost complex structure on S^6

Any such extension is necessarily non-integrable

Abstract

There exists a complex structure $J$ on a connected open subset $S^3_{\delta}\times S^3$ of $S^6$. The present paper proves that: (1) $J$ can be extended to a global almost complex structure $\widetilde{J}$ on $S^6$; (2) any extension to $S^6$ is necessarily non-integrable. Therefore, it is impossible to deform $\widetilde{J}$ to an integrable almost complex structure on $S^6$ while fixing it on $S^3_{\delta}\times S^3$. This phenomenon indicates that the deformation strategy suggested by S.-T. Yau in his Problem 52 cannot be realized in this sense.