Directed immersions for complex structures

Tobias Shin

doi:10.5802/crmath.221

Analyse et géométrie complexes, Géométrie et Topologie

Directed immersions for complex structures

Tobias Shin ¹

¹ Department of Mathematics, SUNY Stony Brook, 100 Nicolls Rd., Stony Brook, NY 11794, USA

Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 773-793.

Résumé

We analyze the differential relation corresponding to integrability of almost complex structures, reformulated as a directed immersion relation by Demailly and Gaussier. Using recent results of Clemente from $2020$ in combination with this analysis, we show the following two statements: first, there are no formal obstructions to integrability of a complex structure, in the sense of $h$ -principle. Second, for an almost complex manifold with arbitrary metric $(X, J, g)$ , and for $ϵ > 0$ , there exists a smooth function $f : X \to ℝ$ and almost complex structure $J^{'}$ on $X$ such that $J$ and $J^{'}$ are $C^{0}$ -close on the graph of $f$ with respect to the extended metric on $X \times ℝ$ , and such that the Nijenhuis tensor of $J^{'}$ on the graph has pointwise sup norm less than $C ϵ$ , where $C$ is a constant depending only on $J$ and $g$ .

Reçu le : 2021-04-26
Révisé le : 2021-05-02
Accepté le : 2021-05-02
Publié le : 2021-09-02

Zbl

DOI : 10.5802/crmath.221

Affiliations des auteurs :

Tobias Shin ¹

¹ Department of Mathematics, SUNY Stony Brook, 100 Nicolls Rd., Stony Brook, NY 11794, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2021__359_7_773_0,
     author = {Tobias Shin},
     title = {Directed immersions for complex structures},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {773--793},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {7},
     year = {2021},
     doi = {10.5802/crmath.221},
     zbl = {07390660},
     language = {en},
}

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Tobias Shin. Directed immersions for complex structures. Comptes Rendus. Mathématique, Volume 359 (2021) no. 7, pp. 773-793. doi : 10.5802/crmath.221. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.221/

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