Supercompactness and the Continuum Function

Given a cardinal κ that is λ-supercompact for some regular cardinal λ ≥ κ and assuming GCH, we show that one can force the continuum function to agree with any function F : [κ, λ] ∩ REG → CARD satisfying ∀α, β ∈ dom(F ) α < cf(F (α)) and α < β =⇒ F (α) ≤ F (β), while preserving the λ-supercompactness of κ from a hypothesis that is of the weakest possible consistency strength, namely, from the hypothesis that there is an elementary embedding j : V → M with critical point κ such that Mλ ⊆ M and j(κ) > F (λ). Our argument extends Woodin’s technique of surgically modifying a generic filter to a new case: Woodin’s key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of j on the ghost coordinates. This work answers a question of Friedman and Honzik [1]. We also discuss an open question involving an analogue of the above result in the case that λ is a singular cardinal.