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Let $V$ be a vector bundle over an algebraic curve $C$, ad assume that that $V \cong V \otimes L$ for some line bundle $L$.

If $V$ is decomposable this is clearly possible, for example take $V \cong \mathcal{O}_C \oplus L$ with $L^2 \cong \mathcal{O}_C$.

Q. Does $V \cong V \otimes L$ imply $L \cong \mathcal{O}_C$ if $V$ is simple?

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  • $\begingroup$ If $L$ has a non-trivial section $s$, then don't we have a map $V \otimes \mathcal{O}_{C} \to V \otimes L$ induced by $Id \otimes s$, which is not multiplication by a constant? By your assumption, this would give an endomorphism of $V$ which is not simply multiplication by a constant. This contradicts simplicity. Since either $L$ or $L^{-1}$ has sections, I think this suffices. Is there a hole in this argument? $\endgroup$ – Benighted Aug 17 at 6:12
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    $\begingroup$ @Benighted: if $V\cong V\otimes L$, $L$ is torsion, so it has no global sections unless it is trivial. $\endgroup$ – abx Aug 17 at 7:19
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    $\begingroup$ Let $L$ be a 2-torsion line bundle on $C$, and $\pi :\tilde{C}\rightarrow C $ the associated étale 2-sheeted covering. Let $V=\pi _*M$ for any line bundle $M$ on $\tilde{C} $. Then $V\otimes L\cong V$; if $M$ is not the pull back of a line bundle $M$ on $C$, it is easy to see that $V$ is simple. $\endgroup$ – abx Aug 17 at 7:48
  • $\begingroup$ Thanks for a nice counterexample. It saved me from wasting time for wrong line of argument. $\endgroup$ – Insong Choe Aug 18 at 3:15
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The answer is no, as shown by the following classical example due to Atiyah.

Take an elliptic curve $E$, choose a point $p \in E$ and consider the unique non-split extension $$0 \longrightarrow \mathcal{O}_E \longrightarrow V_p \longrightarrow \mathcal{O}_E(p) \longrightarrow 0.$$ Then $V_p \simeq V_p \otimes L$ for every $2$-torsion line bundle $L$ on $E$, see Remark p. 35 of [F].

Note that $V_p$ is stable and thus simple, see [F], Theorem 9 p. 89.

References.

[F] R. Friedman: Algebraic surfaces and holomorphic vector bundles, Springer 1998.

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    $\begingroup$ Thanks for nice and detailed information. $\endgroup$ – Insong Choe Aug 18 at 3:16

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