The classical definition of Hilbert modular cuspforms as given in say, Hida's "On $p$-adic Hecke algebras for $\mathrm{GL}\_2$ over totally real fields", defines them as holomorphic functions $f: \mathbb{H}^d \to \mathbb{C}$ satisfying the usual moebius transformation properties with respect to a chosen level subgroup and weight. (Here $\mathbb{H}$ is the usual upperhalf plane, and $d=[F:\mathbb{Q}]$ with $F$ your totally real field).


There are also, some slight variants of this construction (as can also be found in Hida's paper), where one picks some subset $J$ of the set of embeddings of $F$ into $\mathbb{C}$ and then require that these functions be holomorphic in the $J$ components and anti-holomorphic in the $J^c$ components (here ${}^c$ is complement). Let me call them $J$-holomorphic. These show up for example in the Eichler--Shimura isomorphism for Hilbert modular forms.


Now, I know the holomorphic forms can be seen as the global sections of some sheaf on some Hilbert modular variety (a la Katz). But what about these $J$-holomorphic ones? Are they $H^0$s of some sheaf? or maybe $H^k$s for some $k>1$ of the modular sheaf? If its the second option then is $k=|J|$?


Thanks



anti-holomorphic Hilbert modular forms as global sections
Your second guess is spot on: these $J$-holomorphic forms are exactly $H^{k}$ of a sheaf on (a smooth compactification of) the Hilbert modular variety, where $k = |J^c|$. This is an instance of a much more general theory which is largely due to Harris. The canonical reference is


Harris, Michael. [Automorphic forms of $\bar{\partial}$-cohomology type as coherent cohomology classes](https://10.4310/jdg/1214445036). J. Differential Geom. 32 (1990), no. 1, 1--63.


