Let $f: X \to Y$ be a proper morphism of algebraic varieties which is not flat. Then for a line bundle $L$ on $X$, is the vanishing of all higher direct images ($R^i f_* L = 0$ for all $i \geq 0$) equivalent to the vanishing of all $H^i(f^{-1}(s), L|_{f^{-1}(s)})$ for all $i \geq 0$ and all points $s \in Y$?
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