Directed immersions for complex structures

Shin, Tobias

Comptes Rendus. Mathématique, Tome 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 $f:X\to ℝ$ and almost complex structure $J^{\prime }$ on $X$ such that $J$ and $J^{\prime }$ 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 $J^{\prime }$ 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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