Solovay models and forcing extensions

We study the preservation under projective ccc forcing extensions of the property of L(R) being a Solovay model. We prove that this property is preserved by every strongly-Sigma(3)(1) absolutely-coc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for Delta(3)(1) absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets. and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions.