Question.Suppose $\kappa$ is a supercompact cardinal and $\lambda > \kappa$ is measurable (or even larger large cardinal if necessary). Is there a set generic extension of the universe in which $\kappa$ remains supercompact, $\lambda$ is preserved and $cf(\lambda)=\omega?$

**Remark 1.** By Gitik-Shelah indestructibility result, if supercompact cardinal is replaced with strong cardinal, then the answer is yes.

**Remark 2**. If we require that the forcing preserves $\lambda^+,$
then the answer is no, as it is shown by Yair.

**Remark 3**. In A note on sequences witnessing singularity - following Magidor-Sinapova, Gitik has conjectured the following:

Conjecture. Suppose that

$V ⊆ W$ models of ZFC with same ordinals,

$κ$ is a regular cardinal in $V$,

$cof(κ) = ω$ in $W$,

$\aleph_1^V=\aleph_1^W,$

$V, W$ agree about a final segment of cardinals.

Then there is a subclass $V′$ of $V$ which is a model of $ZFC$, agree with $V$ about a final segment of cardinals, and there is a sequence witnessing singularity of $κ$ (in $W$) which is generic over $V′$ for either Namba, Woodin tower or Prikry type forcing.

Assuming this conjecture, it seems quite plausible that the answer to the question might be no in general.

**Edition.** I realized that the question has connection with recent work of Woodin:

**Theorem.** Assume $\kappa$ is an extendible cardinal. If Woodin's $HOD$-conjecture holds, then we can not change the cofinality of some large cardinal $\lambda > \kappa$, preserving the supercompactness of $\kappa,$ by set forcing without collapsing $\lambda.$

forcesomething like this? $\endgroup$ – Asaf Karagila May 7 '16 at 12:04