On the optimality of the HOD dichotomy

In the first part of the manuscript, we establish several consistency results concerning Woodin’s $\text{HOD}$ hypothesis and large cardinals around the level of extendibility. First, we prove that the first extendible cardinal can be the first strongly compact in HOD. We extend a former result of Woodin by showing that under the HOD hypothesis the first extendible cardinal is $C^{(1)}$-supercompact in HOD. We also show that the first cardinal-correct extendible may not be extendible, thus answering a question by Gitman and Osinski \cite[\S9]{GitOsi}. In the second part of the manuscript, we discuss the extent to which weak covering can fail below the first supercompact cardinal $\delta$ in a context where the HOD hypothesis holds. Answering a question of Cummings et al. \cite{CumFriGol}, we show that under the $\text{HOD}$ hypothesis there are many singulars $\kappa <\delta$ where ${\text{cf}}^{\text{HOD}}(\kappa )=\text{cf}(\kappa )$ and ${\kappa }^{+\text{HOD}}={\kappa }^{+}\mathrm{.}$ In contrast, we also show that the $\text{HOD}$ hypothesis is consistent with $\delta$ carrying a club of $\text{HOD}$-regulars cardinals $\kappa$ such that ${\kappa }^{+\text{HOD}}<{\kappa }^{+}$. Finally, we close the manuscript with a discussion about the $\text{HOD}$ hypothesis and $\omega$-strong measurability.