The reference book is S.S.Chern's Complex Manifolds Without Potential Theory.

It's used in integrability condition for an almost complex structure to arise from a complex structure. It's claimed that if the almost complex structure is locally defined by the forms $\theta^k$ of type $(1,0)$ which are linearly independent over the ring of complex-valued smooth functions, then the condition is $d\theta^k\equiv0\mod (\theta^1,\dotsc,\theta^n)$. Here $(\theta^1,\dotsc,\theta^n)$ is the ideal generated by $\theta^1,\dotsc,\theta^n$ in the exterior algebra.

Under the hypothesis that the almost complex structure is real analytic (i.e. the field of endomorphisms $J_x\colon T_xM\to T_xM$ such that $J_x^2=-1_x$, then in some local coordinates, $J_x$ is real-analytic), then it follows from Frobenius's theorem that there exist complex local coordinates $z^k$ such that the form is of type $(1,0)$ are linear combinations of $dz^k$. It's found in Chern's book, pp 17.

I don't know which version of Frobenius's theorem is used. The cotangent bundle, and therefore the exterior algebra of differential forms is complexified (i.e., $\otimes_\mathbb{R}\mathbb C$). The version I know can be found in many canonical textbooks, say Warner's GTM94, or Chern's Lectures on Differential Geometry. It's used for smooth distributions. I wonder the real-analytic version used in the preceding illustration, the idea of the proof of the theorem (it's complexified, therefore the original proof cannot apply directly, I think), and especially where the real-analyticity is really used.

Any help? Thanks!


1 Answer 1


After some search work, I found a more complete proof for the preceding argument in Kobayashi and Nomizu's Foundations of Differential Geometry, Appendix A. The Frobenius theorem used here, is in fact a holomorphic version. See here.


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