Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset?
Thanks in advance.
2026-03-26 19:15:53.1774552553
A coherent sheaf is a vector bundle over subvariety?
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No.
Let $I$ be the ideal sheaf of a point $p$ in $X=\mathbf A^2$. On the complement of $p$, the sheaf $I$ is equal to the line bundle $O_X$. But two line bundles which are isomorphic on an open subset $U \subset X$ whose complement has codimension at least 2 must be isomorphic on $X$.