Corollary 10.16.10.3BC40D26B752 Let $a$ be a nef class on $X$. For every positive-dimensional irreducible reduced subvariety $Y\subset X$,
\[ \mathcal R(a;Y)=\int _ Ya^{\dim Y}. \]
If $\int _ Xa^ n>0$, then
43D7AA1ADEB6
10.16.11
\begin{equation} \bigcup _{\substack {Y\subset X\ \mathrm{irreducible},\ \dim Y>0\\ \mathcal R(a;Y)=0}}Y =\operatorname {Null}(a)=E_{\mathrm{nK}}(a). \end{equation}
Proof.
The pullback of $a$ to a Kähler resolution of $Y$ is nef: pull back smooth representatives with arbitrarily small negative part, and bound the pullback of $\omega $ by a multiple of a Kähler form on the resolution. Since the intersection number $\int _ Ya^{\dim Y}$ is defined through the same pullback, (10.16.6) gives the first assertion. The first equality in (10.16.11) then follows from the definition of the null locus, and the second is the classical theorem on nef and big classes recalled earlier in this chapter.
$\square$
Pome
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