Let $X$ be a complex manifold and $E$ be a holomorphic vector bundle over $X$. Denote $\tilde{E}$ as the pull-back bundle of $E$ over $X\times X$ via the map $\pi: X\times X\to X$ sending $(x,y)\to x$. Assume there exists a global holomorphic section $s:X\times X\to \tilde{E}$ such that $s(x,x)=0$ for all $x\in X$. Set $\triangle=\{(x,y): x=y, x,y\in X\}$ and $A=\{(x,y): s(x,y)=0\}$, so $\triangle\subset A$. If for each fixed $x\in X$, $s(x,\cdot)$ is a local biholomorphism near $y=x$ in the second variable, show that the set $Y=A\backslash \triangle$ is closed in $X\times X$.
The problem is wrong in the topological category as shown by this example: Show the set is closed