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Let $f:X\to S$ be a proper map of complex analytic spaces defined locally by holomorphic functions, with $S$ reduced, and let $\mathcal{F}$ be a coherent analytic sheaf flat over $S$. Ifis locally constant, then the higher direct image sheaf $R^q{f}_{\ast }\mathcal{F}$ is locally free and the canonical comparisonis an isomorphism. Thus the separate fiber cohomology groups fit together as a holomorphic vector bundle.^[ref]

Properness supplies finite-dimensional cohomology and coherent direct images; flatness controls the variation of the coefficients. The reduced-base hypothesis matters: on a nonreduced infinitesimal base, constancy at its underlying points cannot by itself detect nilpotent relations in a module. It must not be substituted for the full base change criterion there.

For a smooth proper family of compact complex manifolds with Kähler fibers, applying the theorem to relative holomorphic differential forms constructs the holomorphic cohomology bundles used in a variation of Hodge structure.