= Cohomology growth for nef twists
{title2=$h^i(X,\mathcal F(mN))=O(m^{d-i})$}
Let $N$ be a <nef Cartier divisor> and let the <coherent sheaf> $\mathcal F$ have support dimension $d$. Then $h^i(X,\mathcal F(mN))=O(m^{d-i})$ for $1\leq i\leq d$. This does not require reducedness or smoothness.
Choose a very ample $A$ and fix $a$ sufficiently large for <Fujita vanishing>. Choose a section of $aA$ avoiding the <associated points> of $\mathcal F$. Its multiplication gives $0\to\mathcal F\to\mathcal F(aA)\to\mathcal Q\to0$, where $\mathcal Q$ has support dimension at most $d-1$. After tensoring with $\mathcal O_X(mN)$, the middle term has zero higher <sheaf cohomology>, uniformly in $m$. Hence $h^i(\mathcal F(mN))\leq h^{i-1}(\mathcal Q(mN))$. Induct on support dimension: for $i=1$ use the <polynomial bound for sections of a fixed divisor>, and for $i>1$ use the induction hypothesis. Degree greater than $d$ vanishes by <Grothendieck vanishing>.
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