Abstract
We study a family of surfaces of general type that arises from the intersections of two translates of the theta divisor on a principally polarized complex abelian fourfold. In particular we determine the N\'eron-Severi lattices of these surfaces and show that for a general abelian fourfold the monodromy group of the associated variation of Hodge structures is a subgroup of index at most two in the Weyl group W(E_6), whereas for a Jacobian variety it degenerates to a subgroup of the symmetric group S_6.
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