On $(\Sigma^2_1)^{uB}$ Absoluteness Between V and HOD

Abstract

We put together Woodin's $\Sigma^2_1$ basis theorem of AD$^+$ and Vop\v{e}nka's theorem to conclude the following: If there is a proper class of Woodin cardinals, then every $(\Sigma^2_1)^{\mbox{uB}}$ statement that is true in $V$ is true in $\mbox{HOD}$. Moreover, this is true even if we allow a parameter $C \subseteq \mathbb{R}$ such that $C$ and its complement have scales that are $\mbox{OD}$ and universally Baire. We also investigate whether $(\Sigma^2_1)^{\mbox{uB}}$ statements are upwards absolute from $\mbox{HOD}$ to $V$ under large cardinal hypotheses, observing that this is true if $\mbox{HOD}$ has a proper class of Woodin cardinals. Finally, we discuss $(\forall^{\mathbb{R}})\, (\Sigma^2_1)^{\mbox{uB}}$ absoluteness and conclude that this much absoluteness between $\mbox{HOD}$ and $V$ cannot be implied by any large cardinal axiom consistent with the axiom ``$V =$ Ultimate $L$''.