Definition of holomorphic differential forms

Question

Reading Forster's Lectures on Riemann surfaces I find his definition of holomorphic differential forms unpleasant. Let $X$ be a Riemann surface. He says that they are forms $\omega$ with value in $T^{1,0}$ (the complex line generated by $\mathrm{d}z$ in local coordinates, or equivalently the eigenbundle of the complex cotangent space $T^*X \otimes \mathbb{C}$ associated to the eigenvalue $i$ of the section $J$ of $\mathrm{End}(T^*X \otimes \mathbb{C}) $ obtained from the complex structure on the real tangent space), such that in local coordinates given by $z$, the function $f$ such that $\omega=f\mathrm{d}z$ be holomorphic on $X$, which is the same as $\omega$ being closed. Now, for instance, for a real $C^k$ manifold $M$, a $C^k$ $p$-form $\omega$ is any section of $\Lambda^p T^*M$ (that is any function of sets $M \to \Lambda^p T^*M$ that maps $\{a\}$ to $T_aM$) such that for any $p$ $C^k$-vector fields $X_1,\dots,X_p$, the resulting function $\omega(X_1,\dots,X_p)$ is $C^k$. Is there an analogous characterization of holomorphic differential forms, perhaps with a complex manifold structure on $T^*X$ ?

Answer 1

The analogous definition you are probably looking for is just as holomorphic sections of $\bigwedge^p T^*X$ (where a holomorphic section is the same thing as a $C^k$ section where we just require the function in question to be holomorphic instead). This definition is just fine and we usually denote these sections $H^0(X, \bigwedge^p T^*X)$. Note that as a matter of fact $ \bigwedge^p T^*X \cong \bigwedge^{p,0} T^*X := \bigwedge^p \left( T^{1,0}\right) ^*X$ as complex vector bundles, so the definitions are indeed equivalent.

You have to be careful when you say "which is the same as ω being closed" thought. This is not true, the form $\bar{f} d\bar{z}$ is closed (On a Riemann surface!) just as well. In general we denote with $\mathcal{A}^{p,q}(X)$ the $smooth$, i.e. $C^\infty$-sections of form $\sum f_I \hspace{0.1cm} dz^{i_1}\wedge ...\wedge dz^{i_p} \wedge d\bar{z}^{j_1} \wedge ... \wedge d\bar{z}^{j_1}$. In this case the elements of $\mathcal{A}^{p,0}(X)$ that are closed under the action of the Dolbeaut-operator $\bar{\partial}$ are precisly the holomorphic forms $H^0(X, \bigwedge^p T^*X)$ and I guess that is what you meant when you said "closed".

Now, I guess the real thing you wanted to know is why we have this asymmetry in the definition of holomorphic forms. When we talk about $C^k$-manifolds we consider $C^k$-sections, so why do we care so much about the smooth sections in the complex case? One answer is, that differential forms can be used to compute the cohomology of manifolds (roughly the shape - how many holes?). The thing is that smooth functions are in some sense way easier to handle since they admit partitions of unity, while holomorphic functions that are determined on an ever so small open set are already uniquely determined everywhere they can be defined. For this reason we can e.g smoothly embbed every $C^\infty$-manifold in $\mathbb{R}^n$ (since we can do so locally), but we can not embbed in a holomorphic fashion every complex manifold into $\mathbb{C}^n$. Similarly in the real case we have the beautiful theory of de Rham which tells us how to compute the cohomology of smooth manifolds. Due to the difference between smooth and holomorphic functions we can not do exactly the same thing in the complex scenario but we should not be afraid to "downgrade" and use only the smooth structure to calculate cohomology which in turn will tell us interesting things about our manifold.