Whenever $M$ is some fine-structural $L$-like model we can prove the implication $V=M\Rightarrow\textsf{GCH}$. For $L$ this is due to Gödel, and for the modern extender models it follows simply by construction. The most recent direction in the inner model programme searches for fine-structural models of the form $\textsf{HOD}^N$ with $N$ some "nice" model (most [recently](http://www.logic.univie.ac.at/~uhlenbrock/MuellerSargsyanHODMnxG.pdf) this includes $N=M\_n(x,g)$ for a Turing cone of reals $x$ and $g$ generic for making its least $N$-inaccessible countable). My question is then whether $V=\textsf{HOD}\Rightarrow\textsf{GCH}$ has been shown to be unprovable in general (modulo large cardinals of course)? Is there some pathological $\textsf{HOD}$ which "obviously" doesn't satisfy $\textsf{GCH}$? Also, I'm not even sure if $\textsf{HOD}^{\textsf{HOD}^N}=\textsf{HOD}^N$ holds for these models, as this (apparently) doesn't hold in general.



Is $V=\textsf{HOD}\not\Rightarrow\textsf{GCH}$ consistent?
Yes, one can produce a model of $ZFC+V=HOD$ in which the $CH$ fails. This should follow from [Consistency results about ordinal definability](https://www.sciencedirect.com/science/article/pii/0003484371900052?via%3Dihub).


In fact, I think the arguments of my paper [HOD, V and the GCH](http://math.ipm.ac.ir/~golshani/Papers/HOD,%20V%20and%20the%20GCH.pdf) can be used to produce a model of $ZFC+V=HOD$ in which the $GCH$ fails everywhere.


Also by a result of Apter (see [Large cardinals need not be large in HOD](http://www.logic.univie.ac.at/~sdf/papers/joint.joel.yong.pdf) for a proof), if there are class many Laver indestructible supercompact cardinals, then $V=HOD$ holds. Note that in such a model $GCH$ must fail at class many regular cardinals.


