Suppose X,Y are two complex manifolds and E,F are vector bundles over them respective, what can I say about their global sections?
Does this formula $\Gamma (X\times Y,p\_1^{\*}E\otimes p\_2^{\*}F)=\Gamma (X,E)\otimes\Gamma (Y,F)$ hold？


If it holds，then we will get every holomorphic function of two complex variables will
be the form $f\_1(z\_1)g\_1(z\_2)+\cdots+f\_n(z\_1)g\_n(z\_2)$. It seems plausibile.



Global sections of tensor product of pull-back of two vector bundles
Perhaps the easiest counterexample comes from letting $X$ and $Y$ be countable infinite disjoint unions of points, and setting the vector bundles to be one-dimensional. Then you just do a dimension count.


A more sophisticated counterexample is the function $e^{xy}$ on $\mathbb{C}^2$. It is globally holomorphic, but it is not a finite sum of products of single-variable entire functions. You can check this by considering the restriction to $\mathbb{N} \times \mathbb{N}$, and writing the values as entries in a big matrix. The matrix is a union of nested van der Monde matrices of nonzero determinant, so it has infinite rank. Therefore, it can't be the sum of finitely many rank 1 matrices of the form $f\_i^T g\_i$


When one of the manifolds is compact, the isomorphism is a corollary of cohomology and base change. You can find a proof in pretty much any book on several complex variables. See, e.g., Grauert, Remmert *Coherent analytic sheaves*.


