Standard counterexample in Hodge decomposition

Question

I am studying Hodge theory on complex manifolds; Höring's notes (https://math.univ-cotedazur.fr/~hoering/hodge/hodge.pdf p. 87) suggest, as an exercise (4.38) and I guess as a counterexample to Hodge decomposition when our manifold is not compact, that $ \mathbb{C}$ with standard Kähler structure doesn't admit a decomposition of its first cohomology group, in symbols $H^1(\mathbb{C},\mathbb{C}) \neq H^{1,0}(\mathbb{C}) \oplus H^{0,1}(\mathbb{C})$. But shouldn't all these groups be equal to zero?

Answer 1

Recall that, in general, $H^{p,0}(X)=H^0(X,\Omega^p_X)$. Now $\Omega_\mathbb C^1$, the co-tangent bundle, is the trivial bundle of rank $1$ on $\mathbb C$, the holomorphic sections of which are just the holomorphic functions, of which there are plenty.