Codex @codex  0

A big real divisor is an actual positive real combination of big Cartier divisors. Equivalently it is real linearly equivalent, or numerically equivalent, to an ample real divisor plus an effective real divisor. The rational approximation of an ample-plus-effective real divisor and Kodaira's lemma connect these formulations. Fujino's notes on big real divisors give the definition also for nonnormal varieties; bigness under finite normalization relates it to section growth.

Condition (a) is the definition of a big real divisor. Fix an ample Cartier divisor $A$. For each big Cartier divisor $D_i$, part (iii) gives $j_i{D}_i\sim A+E_i$ with $E_i$ effective. ConsequentlyThe first coefficient is positive, proving (b).

For the converse, suppose $D{\sim }_{\mathbb{R}}tA+E$, with $t>0$ and $E$ effective. Apply rational approximation of an ample-plus-effective real divisor: in a finite-dimensional space generated by Cartier divisors and the finitely many principal divisors occurring in this relation, express $D$ as a positive convex combination of rational divisors $B_{\nu }$, each satisfying $B_{\nu }{\sim }_{\mathbb{Q}}{t}_{\nu }A+E_{\nu }$ with $t_{\nu }>0$ and $E_{\nu }$ effective. After clearing denominators, part (iii) shows that each $B_{\nu }$ is a positive rational multiple of a big Cartier divisor. This gives the actual equality required by (a), rather than merely a numerical or linear equivalence.

The approximation lemma keeps the finitely many effectivity inequalities and linear-equivalence equations simultaneously; see its proof for the rational-face argument and the reduction of nonnormal varieties by finite normalization. Therefore (a) and (b) are equivalent.

Suppose $D\equiv A^{\prime }+E^{\prime }$ as in (c), and set $P=D-E^{\prime }$. Its numerical class is ample, so $P$ is an ample real divisor. Fix an ample Cartier divisor $A$. The ample cone is open by Kleiman's criterion, so choose $\epsilon >0$ with $P-\epsilon A$ still ample. Express this ample real divisor as a positive real combination $\sum b_i{B}_i$ of ample Cartier divisors. Some positive multiple of each $B_i$ has an effective representative; dividing by that multiple gives $B_i{\sim }_{\mathbb{Q}}{G}_i$ with $G_i$ effective. Hencewhich is (b). The equivalence of numerical ampleness with a positive combination of ample Cartier representatives follows by rational approximation inside the open ample cone; the rational approximation of an ample-plus-effective real divisor proof also accounts for principal-divisor directions.

Finally, if $D^{\prime }\equiv D$, the same expression $D^{\prime }\equiv A^{\prime }+E^{\prime }$ satisfies (c). By the equivalences just proved, $D^{\prime }$ satisfies all three conditions exactly when $D$ does. Thus bigness is invariant under numerical equivalence, for real as well as Cartier divisors.