The $\mathsf{H}\mathsf{O}\mathsf{D}$ Hypothesis and a supercompact cardinal

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