Direct image of coherent analytic sheaves

Question

Given holomorphic varieties $X$ and $Y$, and let $\phi: X \rightarrow Y$ be a morphism between these varieties. Consider a coherent analytic sheaf $\mathcal{S}$ on $X.$ Why, in general the direct image $\phi_{*} \mathcal{S}$ is not a coherent analytic sheaf on $Y$? Could someone give to me a counterexample?

Answer 1

For a very simple example, let $X=\mathbb{C}$, let $Y$ be a point, and let $\mathcal{S}=\mathcal{O}_X$. Then $\phi_*\mathcal{S}(Y)=\mathcal{O}_X(X)$ is the space of all holomorphic functions on $X$. Since this is infinite-dimensional over $\mathcal{O}_Y(Y)=\mathbb{C}$, the sheaf $\phi_*\mathcal{S}$ cannot be coherent.