10.18 Divisorial components at the threshold89CEBB4B3B57

A divisorial component of a Hermitian current consumes part of its Gauduchon trace but contributes no density to its non-pluripolar measure. It can therefore lower the largest coefficient in a Monge–Ampère subsolution inequality. At the maximal coefficient, every weak limit of a maximizing sequence of Gauduchon tests annihilates the class of the divisor, and uniform integrability along the sequence makes the limit a maximizer.

In this section $X$ is again an arbitrary compact complex manifold, now of dimension $n\geq 2$, with the Hermitian form $\omega $. Fix a smooth strictly positive probability volume form $\mathrm{d}V$ and a nonzero function $0\leq f\in L^1(\mathrm{d}V)$, and write $p=f^{1/n}$ for its root density. Let $b=b_\omega (f)$ be its subsolution threshold: by theorem 15.28.4, it is the largest $a\geq 0$ for which some $u\in \mathrm{PSH}(X,\omega )$ satisfies $\mathrm{MA}^{\mathrm{ext}}_\omega (u)\geq af\, \mathrm{d}V$, where $\mathrm{MA}^{\mathrm{ext}}_\omega $ is the extended non-pluripolar Monge–Ampère measure of theorem 13.14.3. Such a $u$ with $a=b$ is a threshold subsolution in the sense of section 15.28. We also use the following notation from section 11.7, section 12.10. The set $\mathscr G_\omega $ consists of the positive $(n-1,n-1)$-currents $T$ with $\mathrm{dd}^{\mathrm{c}}T=0$ and $\int _ X\omega \wedge T=1$, and $\mathscr C_\omega \subset \mathscr G_\omega $ is the weak closure of its smooth strictly positive members, the normalized powers $\gamma ^{n-1}$ of Gauduchon metrics $\gamma $. For $T\in \mathscr G_\omega $, the determinant measure of $T$ has the form $\Gamma _0(T)=k_ T\, \mathrm{d}V$ by theorem 12.9.1, and $D_ p(T)=\int _ Xp k_ T\, \mathrm{d}V$. The root norm $\rho _\omega (p)=b^{-1/n}$ is the supremum of $D_ p$ over $\mathscr G_\omega $, and $\mathcal D_\omega ^0$ is the closure of the bounded functions in the space of roots with finite root norm. Finally, $\kappa _\omega (\beta )$ is the linear positivity threshold of a smooth real $(1,1)$-form $\beta $ from corollary 11.6.2, the infimum over Gauduchon metrics $\gamma $ of $\int _ X\beta \wedge \gamma ^{n-1}/\int _ X\omega \wedge \gamma ^{n-1}$.

Geometrically, at the threshold $c=b$ with $E\ne 0$, the equation $\int _ X\beta _ E\wedge T=0$ defines a nonempty proper exposed face of $\mathscr C_\omega $. Indeed, $\mathscr C_\omega $ is compact and convex, and the pairing with $\beta _ E$ is continuous and nonnegative on it and strictly positive on every smooth strictly positive member. The proposition places every cluster current of a smooth maximizing sequence on this face. The statement concerns $\mathscr C_\omega $, which can be strictly smaller than $\mathscr G_\omega $, as the next example shows. Moreover, for a singular $T$ the smooth pairing $\int _ X\beta _ E\wedge T$ does not define a product $[E]\wedge T$.

The difference between $\mathscr C_\omega $ and $\mathscr G_\omega $ comes from positive currents that pair negatively with a class which is nonnegative on all smooth tests. Integration currents along negative curves give such examples.

Let $m_\omega (\beta )$ be the minimum of $\int _ X\beta \wedge T$ over all $T\in \mathscr G_\omega $, as in section 12.34, whereas by (11.7.2) the threshold $\kappa _\omega (\beta )$ is the minimum over $\mathscr C_\omega $. Since $[C]$ is a positive representative of the class of $\beta _ C$, the two thresholds satisfy

Thus extending the minimum in (11.7.2) to all normalized Gauduchon currents changes its value in this example.

We return to the general setting $(X,\omega ,f)$ of this section. The next two corollaries assume uniform integrability along one smooth maximizing sequence. The condition $p\in \mathcal D_\omega ^0$ suffices, since it gives $b>0$ and uniform integrability over the whole of $\mathscr G_\omega $.

The semipositive criterion and the nef-volume criterion detect different obstructions. On the Hopf threefold, the class of a hyperplane divisor has zero variational volume, so corollary 10.18.5 says nothing about it, but its nowhere vanishing semipositive representative excludes a divisorial component at the threshold.

To see this, let $X$ be the Hopf threefold with its Hopf metric $\omega =\alpha +\zeta $ (section 23.1, section 12.36), where $\alpha $ is the pullback of the Fubini–Study form under the elliptic fibration $X\to \mathbb P^2$, and let $H$ be the inverse image of a projective hyperplane. The class of $H$ is represented by $\alpha $, which is semipositive of rank two everywhere. Its variational volume vanishes for every smooth positive volume form $\mu $: the metrics $\gamma _ t=\alpha +t\zeta $, $t>0$, are Gauduchon by the proof of proposition 12.36.1, and since $\alpha ^3=\zeta ^2=0$ and $\omega ^3=3\alpha ^2\wedge \zeta $, one has $\int _ X\alpha \wedge \gamma _ t^2=\tfrac {2t}{3}\int _ X\omega ^3$ and $\gamma _ t^3=t\, \omega ^3$. Hence the quotient in definition 12.2.1 is $O(t^{1/3})$ as $t\downarrow 0$. By corollary 23.1.40, proposition 15.32.9, one has $\mathcal D_\omega ^0=L^3$ on this example. Hence, for every nonzero nonnegative $L^1$ density, a threshold subsolution has zero generic Lelong coefficient along $H$. Otherwise divisorial removal, as in proposition 12.35.1, would give $\omega +\mathrm{dd}^{\mathrm{c}}u\geq a[H]$ for some $a>0$, contradicting corollary 10.18.4 applied to $aH$. This exclusion uses no bounded mass property. It is also stronger here than nonbigness of the class of $H$: the residual formulas of proposition 12.36.1 allow a positive coefficient along $H$ when the equation constant is smaller than the threshold.

These conditions concern the maximal global coefficient. Whether threshold subsolutions with divisorial components can be constructed from the resulting annihilating currents is open; the threshold within a prescribed singularity type is a different problem.