PomeEA449D56F8F7EA449D56F8F7
We return to arbitrary dimension $n\geq 1$. If $\theta $ is nef and $h$ is a Hermitian form, then the backgrounds $\theta +\epsilon h$ lie in $\mathscr P$ for every $\epsilon >0$, directly from the definition of nefness; their volumes are therefore scalars of smooth equations. As $\epsilon $ decreases to zero, the infimum over Gauduchon metrics commutes with the decreasing limit of the background. Thus the boundary value is determined by smooth equations, even when no metric attains it.
Theorem 12.11.1.9ECB2B21B601 For every nef background $\theta $ and every Hermitian form $h$,
Both quantities on the right decrease as $\epsilon \downarrow 0$, and their limits do not depend on $h$.
Proof. For $0<\epsilon '<\epsilon $ the difference of the two backgrounds is the positive form $(\epsilon -\epsilon ')h$, so the monotonicity in proposition 12.2.2 shows that $\lambda _\mu (\theta +\epsilon h)$ decreases as $\epsilon \downarrow 0$. Its limit is therefore the infimum over $\epsilon >0$. Infima over $\epsilon >0$ and over Gauduchon metrics commute, and for a fixed metric the quotient is affine in $\epsilon $. Hence
Here the second equality holds because the coefficient of $\epsilon $ is positive, so the inner infimum is the value at $\epsilon =0$. This proves the first identity, and its right side is the definition of $\lambda _\mu (\theta )$, which does not involve $h$. The second identity follows by raising the first to the $n$th power, since $\lambda _\mu (\theta +\epsilon h)^ n=c_{\theta +\epsilon h}(\mu )$ by theorem 12.2.3. $\square$
If $X$ is Kähler and $\theta $ is closed and nef, choose $h$ Kähler. Each perturbed solution form is then closed and cohomologous to $\theta +\epsilon h$, so integrating its equation and applying Stokes' theorem gives
Letting $\epsilon \downarrow 0$, we obtain $\mathcal V_\mu (\theta )=\int _ X\theta ^ n$ for every closed nef $\theta $ on a compact Kähler manifold and every $\mu $. We refer to this as the closed nef formula.
Proposition 12.11.2.854AAD5CF1CA Suppose $\theta $ is pseudoeffective and $\lambda _\mu (\theta )=0$. The infimum in definition 12.2.1 is attained if and only if $\theta =\mathrm{dd}^{\mathrm{c}}f$ for a smooth real function $f$.
Proof. Since $D_\mu (\gamma )>0$, a metric $\gamma $ attaining the value zero satisfies $\int _ X\theta \wedge \gamma ^{n-1}=0$. By lemma 11.6.3, including its dimension-one case, $\theta $ is then smoothly $\mathrm{dd}^{\mathrm{c}}$-exact. Conversely, if $\theta =\mathrm{dd}^{\mathrm{c}}f$, Stokes' theorem makes every quotient zero, so every admissible metric attains the infimum. $\square$
For example, let $X$ be the ruled surface and $D$ the section of lemma 16.29.16. The class $c_1(\mathcal O_ X(D))$ is nef, has zero square, and contains the nonzero positive current $[D]$. By the closed nef formula its volume is zero. A smooth $\mathrm{dd}^{\mathrm{c}}$-exact representative would make $[D]$ itself $\mathrm{dd}^{\mathrm{c}}$-exact, which is impossible because $\int _ D\kappa >0$ for a Kähler form $\kappa $. By proposition 12.11.2, no Gauduchon metric minimizes the defining quotient.
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