Local Definability of $\mathsf{HOD}$ in $L(\mathbb{R})$

Abstract

We show that in $L(\mathbb{R})$, assuming large cardinals in $V$, $\mathsf{HOD}\parallel {\eta }^{+\mathsf{HOD}}$ is locally definable from $\mathsf{HOD}\parallel \eta$ for all $\mathsf{HOD}$-cardinals $\eta \in [{\mathbf{\delta }}_1^2,\Theta )$. This is a further elaboration of the statement "${\mathsf{HOD}}^{L(\mathbb{R})}$ is a core model below $\Theta$" made by John Steel.