Let $E$ be an elliptic curve. Let $V$ be a vector bundle, considered as an element of the derived category $D^b(\text{coh }X)$. Is there an autoeqivalence that maps $V$ to a line bundle? Thanks.
autoequivalence of $D^b(\text{coh }X)$ mapping rank $n$ vector bundle to rank $1$ bundle.
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algebraic-geometry
1 Answers
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In general, at least, no.
If $V$ is a trivial vector bundle of rank $n$, then its endomorphism algebra in the derived catgory is non-commutative, and an autoequivalence will preserve that. But a trivial vctor bundle of rank $1$ has a commutative endomorphism algebra in $D(X)$.