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Let $X$ be a connected projective noetherian scheme over $\mathbb{C}$, with every irreducible component of the same dimension. Let $\dim X=n \ge 2$ and $p$ be a closed point on $X$. Denote by $U$ the open subset $X \backslash p$. Let $\mathcal{F}$ be a locally free sheaf on $U$. The question is: for the open immersion $i:U \to X$ is the pushforward $i_*\mathcal{F}$ a locally free sheaf on $X$?

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    $\begingroup$ No, but if $X$ is normal, it will extend to a reflexive sheaf. $\endgroup$ – Karl Schwede Apr 4 '15 at 14:49
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I think that the answer is no in general. For instance, it seems that a locally free sheaf on $U$ does not even systematically extend to a locally free sheaf on $X$.

You can find a counterexample in http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0619.0624.ocr.pdf for $X=\mathbb{C}^3$ and $p=0$. (at the bottom of page 620).

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