# A generalization of the Grauert direct image theorem

EDIT: Let $f\colon X\to Y$ be proper holomorphic submersive map of complex analytic manifolds. Let $\mathcal{F}$ be the sheaf of holomorphic sections of a holomorphic vector bundle over $X$. Assume that for some $i$ the function $y\mapsto dim H^i(X_y,\mathcal{F_y})$ is constant on $Y$. Then the Grauert theorem says that the sheaf $R^if_*\mathcal{F}$ is locally free of finite rank, and that for any point $y\in Y$ the canonical map from the fiber $$(R^if_*\mathcal{F})|_y\to H^i(X_y,\mathcal{F}_y)$$ is an isomorphism, where $X_y:=f^{-1}(y)$, $\mathcal{F}_y$ is the pull-back of $\mathcal{F}$ from $X$ to $X_y$.

Question. Whether the following generalization of the Grauert theorem holds? Under the above assumptions on $X,Y,\mathcal{F}, f$ and $i$, let us fix a point $y\in Y$ and a natural number $n$. Let $y^{(n)}$ be the $n$th infinitesimal neighborhood of $y$, $X_n$ the $n$th infinitesimal neighborhood of the fiber $X_y$, and $\mathcal{F}_n$ be the pull-back of $\mathcal{F}$ to $X_n$. Assume as previously that the function $z\mapsto dim H^i(X_z,\mathcal{F_z})$ is constant on $Y$. We have the canonical map $$R^if_*\mathcal{F}\otimes \mathcal{O}_{y^{(n)}}\to H^i(X_n,\mathcal{F}_n).$$ Is this map an isomorphism? If yes, is there a reference for this fact?

I think this is true in the context when all manifolds and morphisms are algebraic. I think it can be shown by the same method of Ch. III, $\S$ 12 of Hartshorne's "Algebraic Geometry" book. However I need the statement for complex analytic manifolds.

• Your statement is wrong. The pushforward $f_*\mathcal{F}$ might be identically zero, yet $H^0(X_y,\mathcal{F}_y)$ nonzero for some points $y$. Jul 3, 2014 at 12:25
• @JasonStarr: Thanks! I have corrected the statement.
– asv
Jul 3, 2014 at 12:52
• Not exactly what you want, but the only generalisation of the direct image theorem and its applications I know is due to Banica: 1972. I think the same version of the theorem is presented in the book by him and Stanasila, Algebraic Methods in the global theory of complex spaces.
– Ben
Jul 4, 2014 at 7:36
• @BenA.: Thanks for the useful reference. I think the book does contain the positive answer to my question.
– asv
Jul 5, 2014 at 11:17

The statement is explicitly formulated on p. 119 of the book during the proof of Theorem 3.4. It is the claim that the map (**) is an isomorphism, where one should take $Y'=Y$ and $g=Id$.