Let $E$ be a locally free sheaf of rank $>1$ over a smooth projective variety. Then what is $\mathcal{E}xt^1(E,O_X)$? If it's a line bundle, I know that is zero.

Is this $H^1(\mathcal{H}om(E,O_X))$?

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    $\begingroup$ Your notation suggests you want sheaf ext, in which case it is zero. The second paragraph suggests global ext, in which case it may not be zero. $\endgroup$ – Mohan Jun 17 at 13:48
  • $\begingroup$ I want sheaf ext only. Thanks for letting me know that it's zero. $\endgroup$ – user52991 Jun 17 at 13:54

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