Indestructibility and The Level-By-Level Agreement Between Strong Compactness and Supercompactness

Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can. Two important but apparently unrelated results occupy the large cardinal literature. On the one hand, Laver [Lav78] famously proved that any supercompact cardinal κ can be made indestructible by <κ-directed closed forcing. On the other hand, Apter and Shelah [AS97] AMS Math Subject Codes: 03E35, 03E55.