Grauert base change theorem ($(R^q{f}_{\ast }\mathcal{F})\otimes k(s)\cong H^q(X_s,{\mathcal{F}}_s)$)

For a proper holomorphic map and a coherent sheaf flat over the base, local constancy of the fibre-cohomology dimension makes its higher direct image locally free and gives the displayed base-change identification. Applied to relative holomorphic forms in a smooth proper Kähler family, this constructs the holomorphic cohomology bundles entering its variation of Hodge structure.