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Proposition: Let $f:X\rightarrow Y$ be a smooth holomorphic fibration between K\"ahler manifolds, and $L$ be a holomorphic line bundle. Then there exists a Zariski open set $Y_0\subset Y$ such that for any $y\in Y_0$, we have $$f_*(K_{X/Y}+L)_y=H^0(X_y,K_{X_y}+L|_{X_y}), $$ where $X_y=f^{-1}\{y\}$.

I want to whether this statement is true, and if it is, how to prove this.

Another question: Since for any sheaf $\mathcal{A}$ defined on $X$, we know that $f_*(\mathcal{A})_y=H^0(X_y, \mathcal{A}|_{X_y})$ for any $y\in Y$. If we replace $\mathcal{A}$ by $K_{X/Y}+L$, why we could not obtain the formula in the proposition?

Thank you for your attention.

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The answer to the first question is yes, this is the Grauert semi-continuity theorem. It applies to any proper map $f:X\rightarrow Y$ of reduced analytic spaces, and any coherent sheaf $\mathcal{F}$ flat over $Y$. It tells you that there is a Zariski open subset $U$ of $Y$ where $h^0(\mathcal{F}_{|X_y})$ is constant, and this in turn implies $f_*(\mathcal{F})_y=H^0(X_y, \mathcal{F}_{|\,X_y})$ for every $y$ in $U$.

As for the second question, no, we don't know that $f_*(\mathcal{A})_y=H^0(X_y, \mathcal{A}|_{X_y})$ for any sheaf $\mathcal{A}$ on $X$ and any $y\in Y$ — this is definitely false.

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  • $\begingroup$ Thanks for your answer. Grauert's semi-continuity theorem states that $h^0(\mathcal{F}|_{X_y})$ is semi-continuous. How could we find the Zariski open set $U$ of $Y$ such that $h^0(\mathcal{F}|_{X_y})$ is constant in $U$? $\endgroup$
    – Higgs-Boson
    Commented Apr 30, 2015 at 17:34
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    $\begingroup$ An integer-valued function which is upper semi-continuous attains its minimum on an open subset. $\endgroup$
    – abx
    Commented Apr 30, 2015 at 17:37

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