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Proposition 12.20.2883742E77C19

Proposition 12.20.2.883742E77C19 Suppose $X$ is Kähler, $\theta $ is closed and nef, and $h$ is Kähler. Set

\[ r=\max \left\{ 0\leq k\leq n: \int _ X\theta ^ k\wedge h^{n-k}>0\right\} . \]

Then

\[ \underline a(\theta )=\overline a(\theta )=n-r. \]

Thus $n$ minus the common order is the numerical dimension of the closed nef class of $\theta $.

Proof. All the intersection numbers in the definition of $r$ are nonnegative. Indeed, the mixed integrals against $h$ of smooth positive representatives of $\theta +\delta h$ are positive, agree with the cohomological integrals by closed Stokes, and converge to the stated numbers as $\delta \downarrow 0$. The number for $k=0$ is $\int _ Xh^ n>0$, so $r$ is well defined.

By the closed nef formula following theorem 12.11.1,

\[ \mathcal V_\mu (\theta +\epsilon h) =\int _ X(\theta +\epsilon h)^ n =\sum _{k=0}^ n\binom nk\epsilon ^{n-k} \int _ X\theta ^ k\wedge h^{n-k}. \]

As $\epsilon \downarrow 0$, the leading nonzero term is a positive constant times $\epsilon ^{n-r}$, and taking the logarithmic quotient proves the assertion. By proposition 12.20.1, the result holds for every Hermitian perturbation, not only for Kähler ones. $\square$

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