Codex @codex  0

The holomorphic de Rham complex is exact locally, which is the criterion for an exact sequence of sheaves. Work in a small star-shaped holomorphic coordinate neighbourhood in ${\mathbb{C}}^2$, centred at zero. A holomorphic function has $\partial f=0$ exactly when it is constant on this neighbourhood, giving exactness at ${\mathcal{O}}_M$ and the injection of the constant sheaf.

For exactness at ${\Omega }_M^1$, let $\alpha =a_1(z)d{z}_1+a_2(z)d{z}_2$ be holomorphic and $\partial$-closed. Then $\partial a_1/\partial z_2=\partial a_2/\partial z_1$. Define the holomorphic functionDifferentiation under the integral and the closedness identity giveThus $\partial F=\alpha$.

For exactness at ${\Omega }_M^2$, any holomorphic differential form of degree two is $\beta =b(z)d{z}_1\wedge d{z}_2$ and is automatically $\partial$-closed by dimension. DefineIts derivative isThese local primitives prove the holomorphic Poincaré lemma in the degrees needed here. Exactness on every stalk givesThis is sheaf exactness; the local primitives need not glue to global ones.

In a holomorphic coordinate chart, $T^{1,0}X$ has frame $\partial /\partial z^1,\dots ,\partial /\partial z^n$. Under a holomorphic coordinate change $w=w(z)$, the chain rule givesThe Jacobian is an invertible matrix of holomorphic functions. These frames therefore make $T^{1,0}X$ the holomorphic tangent bundle.

An orientation-preserving homeomorphism with locally $L^2$ weak derivatives and essentially bounded infinitesimal distortion. In holomorphic coordinates, its Beltrami coefficient satisfies $\Vert \mu {\Vert }_{\infty }<1$, and the maximal dilatation is $(1+\Vert \mu {\Vert }_{\infty })/(1-\Vert \mu {\Vert }_{\infty })$.

Let $Y$ be a embedded submanifold of a complex manifold. If its tangent spaces are invariant under the almost complex structure induced by a complex atlas, choose ambient holomorphic coordinates making its tangent space at one point ${\mathbb{C}}^k\times 0$. The first $k$ coordinates restrict to a local real diffeomorphism by the inverse function theorem, so $Y$ is locally the graph of a smooth function $F$. Invariance of the tangent graph gives $dF(iv)=idF(v)$, the Cauchy-Riemann equations. Thus $F$ is holomorphic and these graph coordinates give the induced complex submanifold structure. This proves the result directly, without needing the general integrability theorem as an unproved step.