= Algebraic Morse inequality for ample divisors
{title2=$h^0(m(B-C))\ge\frac{B^n-nB^{n-1}C}{n!}m^n+O(m^{n-1})$}
For <ample> rational Cartier classes $B,C$ on an integral projective $n$-fold with $n\ge1$, $B^n>nB^{n-1}\cdot C$ implies $B-C$ is big. Scale to <very ample> integral divisors. Choose an effective <Cartier divisor> $G\in|C|$. Repeated restriction gives $h^0(m(B-C))\ge h^0(mB)-m h^0(G,mB)$: multiply each negatively twisted restriction by a section avoiding its associated points to inject it into $\mathcal O_G(mB)$. <Asymptotic Riemann–Roch> and <Serre vanishing> give the positive leading lower bound along sufficiently divisible section indices after undoing the scaling. No complex-analytic Morse theory or characteristic-zero vanishing is used.
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