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
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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].
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1$\begingroup$ Thank you very much for your answer. Would you know any counter example of the particular case, that is, when a birational map extend to an isomorphism by any chance? $\endgroup$– P. UsadaMay 26 at 6:56
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1$\begingroup$ Let $X = Y$ be a K3 surface and let $C \subset X$ be a smooth rational curve. Then $\mathcal{O}_C$ is a spherical object and the corresponding spherical twist is an autoequivalence of the derived category of $X$, that restricts to the identity over $U = X \setminus C$. $\endgroup$– SashaMay 26 at 8:03