Let $X$ and $Y$ be smooth projective complex varieties. Suppose we have a Fourier-Mukai equivalence
$$
\Phi\_\mathcal P :Perf X \to Perf Y
$$
with kernel $\mathcal P$. Moreover, suppose $\mathcal P$ locally corresponds to a graph of a birational map $\phi:X \supset U \cong V \subset Y$. Now, if there is a (necessarily unique) extension of $\phi$ to a larger open subset $U\subsetneq U'\cong V' \supsetneq V$, is it true that $\mathcal P$ still locally corresponds to the graph of the extended birational map? I am particaularly interested in the case when $\phi$ extends to an isomorphism $X\cong Y$ and wondering if there is a well-known counter example.



A Fourier-Mukai kernel locally given by a graph of a birational map and compatibility with extension
This is not true, you can find a well-known counter example in
[Namikawa, Yoshinori. Mukai flops and derived categories. J. Reine Angew. Math. 560 (2003), 65–76].


