Density of holomorphic sections

Question

Hello!

I am reading an article in which there is the following statement:

Let $E\rightarrow X$ be a holomorphic vector bundle. The holomorphic sections of $E$ over a coordinate neighbourhood of $X$ are dense in the set of smooth sections of $E$.

I have some knowledge in complex geometry but I am not aware of this fact. For which topologies this fact holds? Does somebody knows a place in which I could find some similar statement with a proof? I already had a look in the textbooks of Huybrechts, Demailly and Griffiths-Harris but I did not see a similar statement yet.

I have another question that perhaps is related to the previous one (it is in the same article) but the statement hereafter is from me, not really from the author. Again, let $E\rightarrow X$ be a holomorphic vector bundle. Denote by $\mathcal{E}$ the sheaf of holomorphic sections of $E$. Then we have $\Gamma(U,E)=\mathcal{C}^\infty(U,\mathbb{C})\cdot\mathcal{E}(U)$ with $\Gamma(U,E)$ the sheaf of smooth sections of $E$, $\mathcal{C}^\infty(U,\mathbb{C})$ the sheaf of smooth $\mathbb{C}$-valued functions (seen as real smooth maps I presume) and the dot is just the usual multiplication of sections by functions.

Is this statement true? Does it hold for any holomorphic vector bundle? Where can I find a proof if it is true?

Thank you!

Answer 1

Dear Benjamin, the statement that holomorphic sections are dense in the smooth sections is false, already for the trivial bundle of rank one $E_1=X\times \mathbb C$ over $X=\mathbb C$. Indeed on any non-empty set $U \subset \mathbb C$ it is impossible to approximate the $C^{\infty}$ function $\bar z$ by holomorphic functions since the limit of a sequence of holomorphic functions on $U$ is a holomorphic function on $U$. [The limit is to be understood in the sense of uniform convergence on compact subsets of $U$. There is also an $L^2$- version stating that $L^2(U)\cap \mathcal O(U)$ is a Hilbert subspace of $L^2(U)$, so that a sequence of holomorphic functions converging only in the $L^2$ sense nevertheless has a holomorphic limit. ]

For your "another question", the answer is also "no":

Take $E_2=X\times \mathbb C^2$ , the trivial rank-2 bundle. Then you cannot write the section $(1,\bar z)\in C^{\infty}( U,E_2)$ as $(1,\bar z)=\phi (z) (f(z),g(z))$ with $f,g$ holomorphic and $\phi$ smooth , since $\phi g/\phi f=g/f$ is meromorphic while $\bar z/1=\bar z$ is not ( notice that $\phi$ never vanishes since $1=\phi f$)