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I recently encountered a result about vector bundles on Abelian varieties, which I found interesting. It characterizes homogeneous (translation invariant) vector bundles on Abelian varieties. More precisely any such vector bundle is in the form of $\bigoplus_L L\otimes U_L $, where $U_L$ is a unipotent vector bundle i.e. constructed by successive extensions of trivial line bundle and $L$ is an algebraically trivial line bundle. I wasn't able to find a stand-alone proof of this fact. All proves eventually refer to "M. Miyanishi, Some remarks on algebraic homogeneous vector bundles, in: Number theory, algebraic geometry and commutative algebra, 71–93, Kinokuniya, Tokyo, 1973.". It seems that this reference doesn't exist any more! (at least online). I'd appreciate if anyone knows where to find a proof of this fact.

What I was also interested was understanding the homogeneous sub-bundles of a homogeneous vector bundle. I was wondering whether the proof implies existence of any characterization of such sub-bundles or not? (like does it necessarily imply a homogeneous sub-bundle of $\bigoplus_{L\in A} L\otimes U_L $, is something like $\bigoplus_{L\in B \subseteq A} L\otimes U'_L $, where $U'_L$ is a unipotent sub-bundle of $U_L$ with unipotent quotient. )

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    $\begingroup$ I am pretty sure this result is in some papers of Indranil Biswas. $\endgroup$ – Ben McKay Apr 30 '20 at 19:33
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    $\begingroup$ See Mukai's paper "Duality between $D(X)$ and $D(\hat{X})$ with its application to Picard sheaves". $\endgroup$ – naf May 1 '20 at 1:03
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I don't have access to Miyanishi's article either (at least during lockdown), but as Ulrich suggested, one can look at Mukai's paper "Duality between D(X) and $D(\hat X)$...", where he introduces the Fourier-Mukai transform. I'll denote this transform by $\mathcal{F}$. On page 159, Mukai gives a proof of the characterization of homogeneous bundles you mentioned. Before that in ex 2.9 and 3.2, he proves that

Theorem. $\mathcal{F}$ induces an equivalence between the category of homogenous bundles on an abelian variety $X$, and the category of coherent sheaves on the dual $\hat X$ with finite support. The unipotent bundles correspond to sheaves on $\hat X$ supported at the origin.

Given a homogenous bundle $V$, the points of the support of $\mathcal{F}(V)$ are the line bundles $L$ in your decomposition. The $U_L$ can also be recovered by taking inverse transform of the translate of $\mathcal{F}(V)_L$ back to the origin. Given all of this, it seems clear that homogenous subbundles are as you describe.

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