10.14 Null loci and numerical restricted volumes9CCCA75301B4

In this section $X$ is Kähler and $\omega $ is a Kähler form. For a closed class $\alpha $ and an irreducible analytic subvariety $V\subset X$ of positive dimension, the intersection number $\int _ V\alpha ^{\dim V}$ is computed by pulling back a smooth closed representative to a resolution of $V$; by Stokes' theorem it does not depend on the representative. For a nef class $\alpha $, the subvarieties on which this number vanishes form the classical null locus

For a nef and big class, the classical theorem discussed in section 25.18 gives

The same equality holds on compact manifolds in class $\mathcal C$.

For a pseudoeffective class that is not nef, intersection numbers are replaced by numerical restricted volumes. Let $V\subset X$ be an irreducible compact analytic subvariety of dimension $k>0$, with regular locus $V_{\mathrm{reg}}$. If $V\subset E_{\mathrm{nn}}(\alpha )$, the numerical restricted volume of $\alpha $ along $V$ is defined to be zero. Otherwise, for real $\varepsilon >0$, let $T$ range over the closed currents in $\alpha $ with $T\geq -\varepsilon \omega $ that have analytic singularities with smooth remainder and whose polar set does not contain $V$, and set

where the subscript $\mathrm{ac}$ denotes the absolutely continuous part of a measure on $V_{\mathrm{reg}}$. Because $T$ has smooth remainder, the integrand is the extension by zero of a smooth measure defined away from an analytic subset of $V$.

The numerical null locus of $\alpha $ is the union of the positive-dimensional irreducible subvarieties on which (10.14.1) vanishes. The trace theory of 20.2 The trace operator expresses these restricted volumes as masses of trace currents on $V$; this is why singularity types and restrictions of multiplier ideals enter into the identification of this locus. Such a trace need not equal the pointwise restriction of a singular potential; see section 20.3.