Does **Con(ZFC+ there exists a strongly compact cardinal)** imply **Con(ZFC+ there exists a strongly compact cardinal $\kappa+ 2^\kappa > \kappa^+$)**?



Failure of GCH at a strongly compact cardinal
The answer to the question is yes, as it is proved by Usuba in the paper
[Strongly compact cardinals and the continuum function](https://arxiv.org/abs/1901.05313).


In fact Usuba proves the following more general result:


**Theorem**. Assume $\kappa$ is a strongly compact cardinal. Then there exists a generic extension in which $\kappa$ remains strongly compact and indestructible under adding any number of new subsets to $\kappa$ (Theorem 3.1 of the above paper).


