I'm trying to understand the proof of the holomorphic version of the Frobenius integrability theorem given in p. 51-52 of Voisin's text "Hodge Theory and Complex Algebraic Geometry I".



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> **Statement:** Let $X$ be a complex manifold of complex dimension $n$ and $E \subset T\_X$ an involutive holomorphic subbundle of complex rank $r$. Then $E$ is integrable in the holomorphic sense. Meaning locally there exist holomorphic functions $\varphi : U \to \mathbb{C}^{n-r}$ with $ker(d\varphi\_\*)=E|\_U$.
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The idea of the proof there (If I understand it correctly) goes as follows:


1. Use the real Frobenius theorem for the real part of the distribution $E\_\mathbb{R}$ to get locally functions $\varphi : U \to V \subset \mathbb{R}^{2(n-r)}$ with $ker(d\varphi\_\*)= E\_{\mathbb{R}}|\_U$
2. The almost complex structure $I : T\_{U,\mathbb{R}} \to T\_{U,\mathbb{R}}$ descends to an endomorphism $I : T\_{U,\mathbb{R}}/E\_{\mathbb{R}}|\_U \to T\_{U,\mathbb{R}}/E\_{\mathbb{R}}|\_U$ which itself descends to an almost complex structure on $V$.
3. There exists a complex submanifold transverse to the fibers of $\varphi$ (possibly restricting $U$) which is locally isomorphic to $V$ and whose complex structure agrees with the almost complex structure from (2). In other words there's locally a section of $\varphi$ whose image is a complex manifold whose almost complex structure agrees with the one from (2).
4. Due to (3) we can put a complex structure on $V$ making $\varphi : U \to V$ a holomorphic map


QED.




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There are several things I'm having trouble understanding. 


**Firstly** step (2) isn't elaborated on in the text and it isn't so clear to me why the almost complex structure should descend down to $V$.


**Secondly** given step (3) it's unclear to me why we need step (2). If there's a section whose image is a holomorphic manifold then there's a unique complex structure on $V$ s.t. $\varphi$ is a holomorphic (isn't there?). 


**Thirdly** and perhaps the most crucial is that step (3) seems dangerously circular. Part of the conclusion of the theorem is that $E$ has an involutive complement in $T\_X$ meaning $E|\_U \oplus F|\_U = T\_X|\_U$ for some involutive holomorphic subbundle $F|\_U$. The existence of the section in (3) is stronger than that assertion. It may be that (3) relies on a weaker statement but it's unclear to me how to deduce it from what was done up until now in the book. 


**Fourthly** Nowhere in the proof is it claimed that the almost complex structure on $V$ makes it isomorphic to an open subset of $\mathbb{C}^{n-r}$. It is not claimed that the almost complex structure is the standard one, nor is it obvious to me that it should be from this perspective. 


**Lastly** There's the issue of what the proof uses which I raised in a comment. It doesn't seem very likely to me that one can prove the holomorphic Frobenius from the real Frobenius theorem alone. The real Frobenius is a [consequence of the smooth Poincare lemma](https://math.stackexchange.com/questions/1419718/wanted-a-purely-algebraic-proof-of-the-frobenius-theorem-on-distributions) (closed form is locally exact) therefore a proof of this sort would give the Holomorphic Poincare lemma as a corollary of the smooth Poincare lemma which doesn't sound reasonable to me. The question is therefore what is used in the proof above which goes beyond the smooth Poincare lemma. It most certainly seems to me that this happens somewhere around (3).



Proof of the holomorphic Frobenius theorem in Voisin's book on Hodge theory (Theorem 2.26)
The usual proofs work in the holomorphic category directly. No need for almost complex structures. Take a holomorphic subbundle $V \subset TM$ of the tangent bundle of a complex manifold $M$. In local holomorphic coordinates $x,y$ we can arrange by linear algebra at some point, say at the origin, that $V\_{(0,0)}$ is $dy=A \, dx$ for some matrix $A$. Then we will see that nearby, $V=(dy-f(x,y)dx)$, some $f(x,y)$ holomorphic.
Consider the map $\phi(x,y,v,w)=(x,y,v)$ for $x,v \in \mathbb{C}^p$ and $y,w \in \mathbb{C}^q$. This map is a submersion $\phi \colon TM \to M \times \mathbb{C}^p$. Restricted to $V$, it is (by dimension count) a local biholomorphism, so has an inverse, $w=f(x,y)v$, linear in $v$ since the fibers $V\_{(x,y)}$ are linear. Take each holomorphic vector field $X(x)$ in the $x$-plane, and lift it to a unique holomorphic vector field $\hat{X}(x,y)$ which projects via $(x,y) \mapsto x$ to $X$. This holomorphic vector field is $\hat{X}(x,y)=(X(x),f(x,y)X(x))$, clearly by linear algebra. Because the holomorphic lifts project to the origin vector fields, their flows project to their flows, so their brackets project to their brackets. So if you lift commuting vector fields, like constant ones, you get ones with vertical brackets. If $V$ is bracket closed, you get zero brackets of those lifts. So commuting vector fields lift to commuting vector fields. The flows of the lifts then give holomorphic integral manifolds, by the holomorphic implicit function theorem. There is a unique one through each point $(x,y)=(0,y)$, and then the flow times along the fields together with the initial $y$ value make holomorphic coordinates in which the fields are coordinates translations.


