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.
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.
