# anti-holomorphic Hilbert modular forms as global sections

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

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. J. Differential Geom. 32 (1990), no. 1, 1--63.