Partial near supercompactness

A cardinal κ is nearly θ-supercompact if for every A⊆θ, there exists a transitive M⊨ZFC− closed under <κ sequences with A,κ,θ∈M, a transitive N, and an elementary embedding j:M→N with critical point κ such that j(κ)>θ and j″θ∈N.2, This concept strictly refines the θ-supercompactness hierarchy as every θ-supercompact cardinal is nearly θ-supercompact, and every nearly 2θ<κ-supercompact cardinal κ is θ-supercompact. Moreover, if κ is a θ-supercompact cardinal for some θ such that θ<κ=θ, we can move to a forcing extension preserving all cardinals below θ++ where κ remains θ-supercompact but is not nearly θ+-supercompact. We will also show that if κ is nearly θ-supercompact for some θ⩾2κ such that θ<θ=θ, then there exists a forcing extension preserving all cardinals at or above κ where κ is nearly θ-supercompact but not measurable. These types of large cardinals also come equipped with a nontrivial indestructibility result. A forcing poset is <κ-directed closed if it is γ-directed closed for all γ<κ in the sense of Jech (2003) [13, Def. 21.6]. We will prove that if κ is nearly θ-supercompact for some θ⩾κ such that θ<θ=θ, then there is a forcing extension where its near θ-supercompactness is preserved and indestructible by any further <κ-directed closed θ-c.c. forcing of size at most θ. Finally, these cardinals have high consistency strength. Specifically, we will show that if κ is nearly θ-supercompact for some θ⩾κ+ for which θ<θ=θ, then AD holds in L(R). In particular, if κ is nearly κ+-supercompact and 2κ=κ+, then AD holds in L(R).