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Hi Spiro: I have had much the same difficulties as you, but I now know a modern proof.

At heart, the original proof is an application of the implicit function theorem. More specifically, let $U$ be a polydisk in $C^n$ consider the sequence of Banach manifolds [ Diff^{k,\alpha}(U,C^n) $Diff^{k,\alpha}(U,C^n) \to AC^{k-1,\alpha}(U) \to (A^{0,2})^{k-2,\alpha}(U,TU). ]A^{0,2})^{k-2,\alpha}(U,TU).$

These are respectively the diffeomorphisms $U\to C^n$ of class $(k,\alpha)$, the almost complex structures on $U$ of class $(k-1,\alpha)$ and the $(0,2)$ forms on $U$ with values in the holomorphic tangent bundle, of class $(k-2,\alpha)$. The first map is the pullback of the standard complex structure, and the second is the Frobenius integrability form [ \phi $\phi \mapsto \overline \partial \phi - \frac 12 [\phi\wedge \phi]. ]phi].$

The object is to show that the first map is locally surjective onto the inverse image of $0$ by the second. These spaces are Banach manifolds, and if you can show that the sequence of derivatives (respectively at the identity, at the standard complex structure and at 0) is split exact, the result follows from the implicit function theorem.

This sequence of derivatives is the Dolbeault sequence on $U$ (in the appropriate class), and it is split exact, though this is NOT obvious. There is an error in the original paper, or rather in the paper of Chern's that it depends on, but the result is true. The remainder of the mess in the original proof is due to the authors writing out the Picard iteration in the specific case, rather than isolating the needed result.

I am working on getting this written up with Milena PibiniakPabiniak, a graduate student here at Cornell. Write me at jhh8@cornell.edu if you are interested in seeing details.

John Hubbard
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Hi Spiro: I have had much the same difficulties as you, but I now know a modern proof.

At heart, the original proof is an application of the implicit function theorem. More specifically, let $U$ be a polydisk in $C^n$ consider the sequence of Banach manifolds [ Diff^{k,\alpha}(U,C^n) \to AC^{k-1,\alpha}(U) \to (A^{0,2})^{k-2,\alpha}(U,TU). ]

These are respectively the diffeomorphisms $U\to C^n$ of class $(k,\alpha)$, the almost complex structures on $U$ of class $(k-1,\alpha)$ and the $(0,2)$ forms on $U$ with values in the holomorphic tangent bundle, of class $(k-2,\alpha)$. The first map is the pullback of the standard complex structure, and the Frobenius integrability form [ \phi \mapsto \overline \partial \phi - \frac 12 [\phi\wedge \phi]. ]

The object is to show that the first map is locally surjective onto the inverse image of $0$ by the second. These spaces are Banach manifolds, and if you can show that the sequence of derivatives (respectively at the identity, at the standard complex structure and at 0) is split exact, the result follows from the implicit function theorem.

This sequence of derivatives is the Dolbeault sequence on $U$ (in the appropriate class), and it is split exact, though this is NOT obvious. There is an error in the original paper, or rather in the paper of Chern's that it depends on, but the result is true. The remainder of the mess in the original proof is due to the authors writing out the Picard iteration in the specific case, rather than isolating the needed result.

I am working on getting this written up with Milena Pibiniak, a graduate student here at Cornell. Write me at jhh8@cornell.edu if you are interested in seeing details.

John Hubbard