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-Σ31 absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for δ31 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.
"Solovay models and forcing extensions." J. Symbolic Logic 69 (3) 742 - 766, September 2004. https://doi.org/10.2178/jsl/1096901764