Determinacy and Woodin cardinals

Question

I am looking for a reference for the following result:

Theorem Assume $\kappa$ is a Woodin cardinal. Then after forcing with the Levy collapse $\mathbb{P}=Col(\omega, \kappa)$, the $\Sigma^1_2$-determinacy holds in $V[G_{\mathbb{P}}]$.

Note that it suffices to show that $\Sigma^1_2$-determinacy holds in $L(\mathbb{R})^{V[G_{\mathbb{P}}]}.$

Answer 1

The result is true if you actually collapse the Woodin cardinal, not just the cardinals smaller than it. This follows from the results in Itay's chapter in the Handbook. See MR2768701, zbM1198.03057

Neeman, Itay. Determinacy in $L(\mathbb R)$. In "Handbook of set theory. Vols. 1, 2, 3", 1877–1950. Springer, Dordrecht, 2010 ISBN:978-1-4020-4843-2

Particularly, see Corollary 6.12 and, really, Chapter 6, which "localizes" the results of Chapter 5 (which, in turn, assume that there is a measurable cardinal above the Woodin, and show that in $V$ we have $\mathbf\Sigma^1_2$-determinacy). The result there is stated with $\Delta^1_2$-determinacy in the conclusion, but Martin proved that if $\mathsf{DC}$ holds, then $\Delta^1_2$-determinacy gives $\Sigma^1_2$-determinacy. (Note these are lightface results). In turn, Martin's theorem is Theorem 6.3 in the Handbook chapter by Peter and Hugh, see MR2768702, zbM1198.03072

Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In "Handbook of set theory. Vols. 1, 2, 3", 1951–2119, Springer, Dordrecht, 2010.

Answer 2

(This answers what was the original question, which was with $\mathrm{Coll}(\omega,{<\kappa})$ replacing $\mathrm{Coll}(\omega,\kappa)$.)

Hmm...doesn't this contradict Theorem 1.22 of "MICE WITH FINITELY MANY WOODIN CARDINALS FROM OPTIMAL DETERMINACY HYPOTHESES" by Mueller, Schindler, Woodin? According to that result, under ZFC, $M_1^\#$ exists and is $\omega_1$-iterable iff boldface-$\Pi^1_1$ and lightface-$\Delta^1_2$ determinacy holds. (So work in $V=M_1$, and let $\delta$ be the unique Woodin cardinal. Let $G$ be $(V,\mathbb{P})$-generic where $\mathbb{P}=\mathrm{Coll}(\omega,{<\delta})$. Suppose lightface-$\Sigma^1_2$ determinacy holds in $V[G]$. Then lightface-$\Delta^1_2$ determinacy holds there. But also, every real has a sharp in $V[G]$, so boldface-$\Pi^1_1$ determinacy holds there also. So $V[G]\models$"$M_1^\#$ exists and is $\omega_1$-iterable". But then $(M_1^\#)^{V[G]}\in\mathrm{HOD}^{V[G]}$, so by homogeneity of the collapse, $(M_1^\#)^{V[G]}\in V=M_1$. But then $M_1\models$"$M_1^\#$ exists and is $\delta$-iterable", which is impossible.)