# Constructing certain Global section with prescribed zero locus over Stein manifold

Let $$X^n$$ be a Stein manifold (complex submanifold in $$\mathbb{C}^N$$ for some large $$N$$). Let $$D = \{(z,z)\in X\times X: z\in X\}$$ be the diagonal in $$X\times X$$. I'm looking for some holomorphic vector bundle $$B$$ of dimension $$n$$ over $$X\times X$$ and a global section $$s : X\times X \to B$$ such that $$D = s^{-1}(0)$$.

There's a construction by Henkin-Leiterer as follows: Take $$B = \pi^* TX$$ where $$\pi: X \times X \to X$$ is the projection onto its first variable and $$\pi^*$$ is the pull-back. Let $$F : X\to \mathbb{C}^N$$ be an embedding for Stein manifold $$X$$. Then $$d F : T(X)\to X\times \mathbb{C}^N$$ is an injection between holomorphic vector bundles. Use Cartan B, one can show that there's a holomorphic left inverse $$(dF)^{-1}: X\times \mathbb{C}^N \to T(X)$$ such that $$(dF)^{-1} (dF)= Id$$. Let $$(z,\zeta)\in X \times X$$, define $$s(z,\zeta)= (\pi)^*(d F)^{-1} (z, F(z)- F(\zeta))$$. One sees that $$D\subset s^{-1}(0)$$. Henkin and Leiterer showed that the extra zero locus (if exists) is disjoint from D by proving that for fixed $$z\in X$$, $$s(z,\zeta)$$ is locally biholomorphism in the second variable.

My question is, can one show that there's no extra zero locus? If not, does there exist such pairing $$(B,s)$$ such that $$s^{-1}(0)=D$$ without extra zero locus? Thanks!