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 $\alpha=a_1(z)\,dz_1+a_2(z)\,dz_2$ be holomorphic and $\partial$-closed. Then $\partial a_1/\partial z_2=\partial a_2/\partial z_1$. Define the <holomorphic function>
$$
F(z)=\int_0^1\bigl(z_1a_1(tz)+z_2a_2(tz)\bigr)\,dt.
$$
Differentiation under the integral and the closedness identity give
$$
\frac{\partial F}{\partial z_j}
=\int_0^1\left(a_j(tz)+t\sum_i z_i\frac{\partial a_j}{\partial z_i}(tz)\right)dt
=[t\,a_j(tz)]_{t=0}^{t=1}=a_j(z).
$$
Thus $\partial F=\alpha$.

For exactness at $\Omega_M^2$, any <holomorphic differential form> of degree two is $\beta=b(z)\,dz_1\wedge dz_2$ and is automatically $\partial$-closed by dimension. Define
$$
\gamma(z)=\int_0^1 t\,b(tz)(z_1\,dz_2-z_2\,dz_1)\,dt.
$$
Its derivative is
$$
\partial\gamma
=\int_0^1\left(2t\,b(tz)+t^2\sum_i z_i\frac{\partial b}{\partial z_i}(tz)\right)dt\,dz_1\wedge dz_2
=[t^2b(tz)]_0^1\,dz_1\wedge dz_2=\beta.
$$
These local primitives prove the <holomorphic Poincaré lemma> in the degrees needed here. Exactness on every stalk gives
$$
\boxed{0\longrightarrow\underline{\mathbb C}\longrightarrow\mathcal O_M
\xrightarrow{\partial}\Omega_M^1\xrightarrow{\partial}\Omega_M^2\longrightarrow0.}
$$
This is sheaf exactness; the local primitives need not glue to global ones.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 118 / 5 / b / Solution

An <exact sequence> of <sheaves of abelian groups> is a sequence of sheaf homomorphisms for which the image of each map equals the kernel of the next. Exactness can be checked on every <stalk of a sheaf>. Exactness of the associated sequences of sections on every open set is a stronger condition: a surjective sheaf homomorphism need only admit lifts locally.

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 Incoming links: Exact sequence of sheaves