= Real Nakai–Moishezon criterion
{title2=$D^{\dim V}\cdot[V]>0\quad\text{for all }\dim V>0$}
A real Cartier class on a <projective scheme> is <ample> if and only if its top self-intersection on every positive-dimensional integral <subvariety> is positive. For the converse curve tests give nefness. Small <ample> perturbations and rational approximation give <ample> rational $B,C$ with $D-A=B-C$ satisfying the <algebraic Morse inequality for ample divisors>, hence bigness. Induction gives <ample> restrictions on codimension-one <subvarieties>; <uniform ample subtraction from a big divisor with ample exceptional restrictions> makes $D-\epsilon A$ <nef>. The <nef-plus-ample ampleness lemma> concludes. For reducible schemes perform the finite component tests simultaneously.
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