  This paper explores several topics related to Woodin's HOD conjecture. We
improve the large cardinal hypothesis of Woodin's HOD dichotomy theorem from an
extendible cardinal to a strongly compact cardinal. We show that assuming there
is a strongly compact cardinal and the HOD hypothesis holds, there is no
elementary embedding from HOD to HOD, settling a question of Woodin. We show
that the HOD hypothesis is equivalent to a uniqueness property of elementary
embeddings of levels of the cumulative hierarchy. We prove that the HOD
hypothesis holds if and only if every regular cardinal above the first strongly
compact cardinal carries an ordinal definable omega-Jonsson algebra. We show
that if the HOD hypothesis holds and HOD satisfies the Ultrapower Axiom, then
every supercompact cardinal is supercompact in HOD.
