Collapsing the cardinals of $HOD$

Abstract

Assuming that $GCH$ holds and $\kappa$ is ${\kappa }^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $\kappa$ remains strongly inaccessible and $({\alpha }^{+}{)}^{HOD}<{\alpha }^{+}$ for every infinite cardinal $\alpha <\kappa$. In particular the rank-initial segment $W_{\kappa }$ is a model of ZFC in which $({\alpha }^{+}{)}^{HOD}<{\alpha }^{+}$ for every infinite cardinal $\alpha$.