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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.

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  • $\begingroup$ What does it mean that a vector bundle splits? $\endgroup$ Jul 4, 2019 at 23:29
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    $\begingroup$ @MikhailBorovoi, a direct sum of line bundles? $\endgroup$
    – LSpice
    Jul 4, 2019 at 23:45
  • $\begingroup$ @MikhailBorovoi Indeed. $\endgroup$
    – jj_p
    Jul 5, 2019 at 13:06
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    $\begingroup$ See the proof of Lemma 1.1 here: arxiv.org/pdf/math/9911192.pdf $\endgroup$ Jul 5, 2019 at 22:35
  • $\begingroup$ @WalterNeff Thanks $\endgroup$
    – jj_p
    Jul 6, 2019 at 17:24

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