Comptes Rendus
Analyse et géométrie complexes, Géométrie et Topologie
Directed immersions for complex structures
Tobias Shin1
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.

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→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×ℝ, and such that the Nijenhuis tensor of J ′ on the graph has pointwise sup norm less than Cϵ, where C is a constant depending only on J and g.

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Révisé le :
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DOI : 10.5802/crmath.221
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 
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     title = {Directed immersions for complex structures},
     journal = {Comptes Rendus. Math\'ematique},
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     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
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     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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