Non-extendability of complex structures

TL;DR

The paper shows that a Calabi–Eckmann type almost complex structure $J$ on the open subset $S^3_\delta\times S^3\subset S^6$ can be extended to a global almost complex structure $\widetilde{J}$ on $S^6$, but any such extension is non-integrable, blocking a deformation to an integrable structure while preserving $\widetilde{J}$. The extension relies on a decomposition of $S^6$ into two disc bundles via isoparametric foliation and a careful homotopy analysis of the coordinating frame $G$, replacing $J_0$ by a convenient homotopy $K$ and extending $G^{-1}J_0G$. The non-integrability result leverages Campana–Demailly–Peternell and Krasnov’s theorems to rule out a complex structure on any compact extension, using the existence of infinitely many Hopf-type hypersurfaces in $S^3_\delta\times S^3$. The final proposition uses obstruction theory and deformation results to justify the constructed almost complex structure and its incompatibility with a global integrable extension, illustrating limitations of Yau’s Problem 52 approach in this setting.

Abstract

There exists a complex structure $J$ on a connected open subset $S^3_δ\times S^3$ 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 $S^3_δ\times S^3$. This phenomenon indicates that the deformation strategy suggested by S.-T. Yau in his Problem 52 cannot be realized in this sense.

Theorems & Definitions (13)

• Theorem 1.1
• Remark 1.1
• Proposition 1.1
• Lemma 2.1
• proof
• Corollary 2.1
• Lemma 2.2
• Corollary 2.2
• Lemma 2.3
• proof