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We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field and prove that every linear quotient of the fundamental group of such a variety is virtually abelian, as well as that its Albanese map is surjective, has connected fibres, and has no multiple fibres in codimension one.

Albanese Maps and Fundamental Groups of Varieties With Many Rational Points Over Function Fields