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!
