Beyond $\text{underTilde}{\mathrm{\Sigma }}_1^2$ absoluteness

There have been many generalizations of Shoenfield’s Theorem on the absoluteness of ${\mathrm{\Sigma }}_2^1$ sentences between uncountable transitive models of $\mathrm{Z}\mathrm{F}\mathrm{C}$. One of the strongest versions currently known deals with ${\mathrm{\Sigma }}_1^2$ absoluteness conditioned on $\mathrm{C}\mathrm{H}$. For a variety of reasons, from the study of inner models and from simply combinatorial set theory, the question of whether conditional ${\mathrm{\Sigma }}_2^2$ absoluteness is possible at all, and if so, what large cardinal assumptions are involved and what sentence(s) might play the role of $\mathrm{C}\mathrm{H}$, are fundamental questions. This article investigates the possiblities for ${\mathrm{\Sigma }}_2^2$ absoluteness by extending the connections between determinacy hypotheses and absoluteness hypotheses.