Is it true that a vector bundle over a toric variety is also a toric variety if and only if it splits? if so, how do we prove it?

This seems to be the content of a remark in Oda's Tata's lectures on torus embeddings, although the language is slightly different, and there's no proof.

  • $\begingroup$ What does it mean that a vector bundle splits? $\endgroup$ – Mikhail Borovoi Jul 4 at 23:29
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    $\begingroup$ @MikhailBorovoi, a direct sum of line bundles? $\endgroup$ – LSpice Jul 4 at 23:45
  • $\begingroup$ @MikhailBorovoi Indeed. $\endgroup$ – jj_p Jul 5 at 13:06
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    $\begingroup$ See the proof of Lemma 1.1 here: arxiv.org/pdf/math/9911192.pdf $\endgroup$ – Walter Neff Jul 5 at 22:35
  • $\begingroup$ @WalterNeff Thanks $\endgroup$ – jj_p Jul 6 at 17:24

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