  We study $p$-adic properties of the coherent cohomology of some automorphic
sheaves on the Hilbert modular variety $X$ for a totally real field $F$ in the
case where the prime $p$ is totally split in $F$. More precisely, we develop
higher Hida theory \`{a} la Pilloni, constructing, for $0\leq q\leq
[F:\mathbb{Q}]$, some modules $M^q$ which $p$-adically interpolate the ordinary
part of the cohomology groups $H^q(X, \underline{\omega}^{\kappa})$, varying
the weight $\kappa$ of the automorphic sheaf.
