In this paper, we prove that: if $\kappa$ is supercompact and the $\mathsf{ HOD}$ Hypothesis holds, then there is a proper class of regular cardinals in $V_{ \kappa}$ which are measurable in $\mathsf{ HOD}$. Woodin also proved this result. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the $\mathsf{ HOD}$ Hypothesis and supercompact cardinals, large cardinals in $\mathsf{ V}$ are reflected to be large cardinals in $\mathsf{ HOD}$ in a local way, and reveals the huge difference between $\mathsf{ HOD}$-supercompact cardinals and supercompact cardinals under the $\mathsf{ HOD}$ Hypothesis.
Categories: math.LO