TITLE: Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness
QUESTION [6 upvotes]: It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ where $\kappa$ is a $\Sigma_2$-correct cardinal.
My questions are about what large cardinal principles can prove $\Sigma_3^1$-generic absoluteness. In particular:
1) If $0^\sharp$ exists (or even $x^\sharp$ exists for all reals $x$), does (light-face) $\Sigma_3^1$-generic absoluteness holds. 
2) What if there is a measurable cardinal, then does $\Sigma_3^1$-generic absoluteness hold?
Is there any large cardinal whose existence implies $\Sigma_3^1$ generic-absoluteness?

REPLY [7 votes]: Re a proof of this result: I once wrote up an elementary proof of this which avoids the Martin-Solovay tree: See Theorem 1 of  https://ivv5hpp.uni-muenster.de/u/rds/talks-barcelona.dvi