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

A holomorphic differential form $\alpha$ of type $(p,0)$ satisfies $\bar{\partial }\alpha =0$. The adjoint ${\bar{\partial }}^{\ast }$ lowers antiholomorphic degree, so ${\bar{\partial }}^{\ast }\alpha =0$ as well. Therefore ${\Delta }_{\bar{\partial }}\alpha =0$.

The Kähler Laplacian identity gives ${\Delta }_d\alpha =2{\Delta }_{\bar{\partial }}\alpha =0$. Compactness lets us use the $L^2$ identity for the Hodge Laplacian:Both terms are nonnegative, soThis proves that holomorphic forms on a compact Kähler manifold are closed. The original PDF uses ${\Omega }_M^p$ and ${\mathcal{A}}_{M,\mathbb{C}}^{p+1}$; the extra symbols in the local TeX superscripts are transcription errors.

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.

For degree zero, compactness and the maximum modulus principle make holomorphic functions constant on each connected component. For degree two, a holomorphic differential form is $\bar{\partial }$-closed and its $\partial$ derivative has type $(3,0)$, which vanishes on a complex surface. Degrees above two have no nonzero holomorphic forms. It remains to treat a holomorphic differential form $\alpha$ of degree one.

Set $\beta =d\alpha =\partial \alpha$. Its coefficients are holomorphic, so $\beta$ is a holomorphic differential form of degree two, hence closed by the degree-two observation. Its conjugate is closed too. The Stokes theorem givesThe integrand is nonnegative in the complex orientation. In local coordinates, if $\beta =b(z)d{z}_1\wedge d{z}_2$ with $z_j=x_j+i{y}_j$, thenIt is strictly positive wherever $\beta$ is nonzero. Its zero integral therefore forces $\beta =0$ everywhere. Thus holomorphic forms on a compact complex surface are closed, even when the surface is not Kähler:

Let ${\mathcal{Z}}_M^1$ be the sheaf of closed holomorphic one-forms, namely $\ker (\partial :{\Omega }_M^1\to {\Omega }_M^2)$. The local exactness proved in part (b) gives the short exact sequence of sheavesIts long exact sequence in sheaf cohomology beginsThe first arrow is an isomorphism: on a compact complex manifold, a global holomorphic function is constant on each connected component, just as a global section of the constant sheaf is. Thus $\delta$ is injective. By part (c), every global holomorphic differential form of degree one is closed, so $H^0(M,{\mathcal{Z}}_M^1)=H^0(M,{\Omega }_M^1)$.

Concretely, for a holomorphic differential form of degree one choose local holomorphic primitives $f_i$. The connecting class $\delta (\alpha )$ is represented by the locally constant Čech cocycle $f_j-f_i$ on overlaps. The following arrow is induced by the inclusion of the constant sheaf into the sheaf of holomorphic functions. Exactness is inherited from the long exact sequence, giving