An elementary proof of the irreducibility of the rank \(g\) Picard bundle (Q1360217)

A (very) general principally polarized complex abelian variety \((A,\theta)\) satisfies Mattuck's property: its algebraic cohomology in degree \(2k\) is generated by \(\theta^k/k!\) [\textit{A. Mattuck}, Proc. Am. Math. Soc. 9, 88-98 (1958; Zbl 0118.37301)]. It follows from a result of Schur that the Chern polynomial of a vector bundle \(E\) on \(A\) with top Chern class \(\pm\theta^k/k!\) is irreducible over \({\mathbb{Q}}\); in particular, \(E\) is irreducible (i.e. is not extension of two vector bundles of smaller ranks). A general Jacobian is proved to satisfy Mattuck's property [this also follows from papers by \textit{S. Mori}, cf. Jap. J. Math., New Ser. 2, 109-130 (1976; Zbl 0339.14016) and 3, 105-109 (1977; Zbl 0379.14011)] hence the general rank \(g\) Picard bundle is irreducible.