Indestructibility properties of remarkable cardinals

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of L(R) is absolute for proper forcing [Sch00]. Here, we study the indestructibility properties of remarkable cardinals. We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestructible by all <κ-closed ≤κ-distributive forcing and all twostep iterations of the form Add(κ, θ) ∗ Ṙ, where Ṙ is forced to be <κ-closed and ≤κ-distributive. In the process, we introduce the notion of a remarkable Laver function and show that every remarkable cardinal carries such a function. We also show that remarkability is preserved by the canonical forcing of the GCH.