Principal Bundles whose Restrictions to a Curve are Isomorphic
Let 𝑋 be a normal projective variety defined over an algebraically closed field 𝑘. Let $|O_X(1)|$ be a very ample invertible sheaf on 𝑋. Let 𝐺 be an affine algebraic group defined over 𝑘. Let $E_G$ and $F_G$ be two principal 𝐺-bundles on 𝑋. Then there exists an integer $n \gg 0$ (depending on $E_G$ and $F_G$) such that if the restrictions of $E_G$ and $F_G$ to a curve $C\in |O_X(n)|$ are isomorphic, then they are isomorphic on all of 𝑋.
Volume 130, 2020
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