Universally Baire Sets and Generic Absoluteness

We prove several equivalences and relative consistency results involving notions of generic absoluteness beyond Woodin’s (Σ ̃ 1)λ generic absoluteness for a limit of Woodin cardinals λ. In particular, we prove that two-step ∃R(Π ̃ 1)λ generic absoluteness below a measurable cardinal that is a limit of Woodin cardinals has high consistency strength, and that it is equivalent with the existence of trees for (Π1) uBλ formulas. The construction of these trees uses a general method for building an absolute complement for a given tree T assuming many “failures of covering” for the models L(T, Vα) below a measurable cardinal.