Did you already try complete intersections of theta divisors in a moduli space of principal $G$-bundles ($G semisimple) on a smooth, projective curve? The weight 1 Hodge structure of the curve induces a weight 3 Hodge structure of the moduli space, which maps to the weight 3 Hodge structure of the complete intersection.
– Jason StarrMay 17 '11 at 20:54