11.41 Divisorial types and moving sections2D3AC831EDA6

Proof. The Poincaré–Lelong formula gives $\theta +\mathrm{dd}^{\mathrm{c}}\varphi =[\Gamma ]+\beta $, so every function on the right is $\theta $-plurisubharmonic and has the type of $\varphi $. Conversely, let $u\in \mathrm{PSH}(X,\theta )$ with $u\sim \varphi $. Then $u$ has the same generic Lelong numbers as $\varphi $ along the components of $\Gamma $, namely the coefficients of $\Gamma $, and lemma 5.38.1 gives

The locally integrable difference $w=u-\varphi $ therefore has a $\beta $-plurisubharmonic representative. It is bounded off the divisor because $u\sim \varphi $, and the local submean property preserves both bounds across the divisor. Since $u$ and $\varphi +w$ agree almost everywhere, they are equal as quasi-plurisubharmonic functions.

Off the divisor, the currents $\theta +\mathrm{dd}^{\mathrm{c}}(\varphi +w)$ and $\beta +\mathrm{dd}^{\mathrm{c}}w$ agree, and locality identifies their Monge–Ampère products there. The extended measure on the left gives the divisor zero mass, and so does the bounded product on the right, which is finite and charges no analytic set by section 5.28. This proves (11.41.2). In the background $\beta $, the equation with bounded density has a continuous solution and a unique constant by theorem 14.19.2, proposition 14.19.34. The density $\mathrm{e}^ F$ is bounded above and below by positive constants, so corollary 14.24.17 gives uniqueness modulo constants. The correspondence transfers these assertions to the whole ordinary type and identifies the functionals. $\square$

Proof. The convergence of the residual solutions follows from proposition 11.40.4 in the first case and from the preceding compactness argument in the second. The fixed metrics and bounded weights give a common smooth lower bound for $\mathrm{dd}^{\mathrm{c}}\varphi _ j$. The convergent sections give a common upper bound for $\varphi _ j$ and convergence at a point outside all limiting divisors, so the suprema of the $\varphi _ j$ are bounded above and below. By theorem 5.25.2, every subsequence has a further subsequence converging in $L^1$. Off the limiting divisors, convergence of the sections and weights gives local uniform convergence to $\varphi $; since those divisors have zero volume, every $L^1$ cluster point equals $\varphi $. Thus $\varphi _ j\to \varphi $ in $L^1(X)$.

Put $C_ j=\sup _ X(\varphi _ j+v_ j)$ and $C=\sup _ X(\varphi +v)$. The sums converge in $L^1$ and have a common smooth lower bound for their complex Hessians, since $\mathrm{dd}^{\mathrm{c}}v_ j\geq -\beta _ j$ and the residual forms are uniformly bounded. Applying lemma 5.25.1 at maximum points gives $\limsup _ jC_ j\leq C$. A maximum point of $\varphi +v$ lies off the poles of $\varphi $, and evaluation there, using local uniform convergence, gives $\liminf _ jC_ j\geq C$.

It follows that $u_ j=\varphi _ j+v_ j-C_ j\to \varphi +v-C=u$ in $L^1$. For an empty family of divisors, the assertion reduces to the convergence of the residual solutions and of their suprema. $\square$

Proof. If $u\preceq \varphi $, its generic Lelong numbers dominate the coefficients of $\Gamma $. By lemma 5.38.1, $\theta +\mathrm{dd}^{\mathrm{c}}u-[\Gamma ]\ge 0$. The locally integrable distribution $w=u-\varphi $ therefore has a canonical $\beta $-psh representative. Off $S$ it equals the difference and is bounded above. The identity $u=\varphi +w$ holds almost everywhere, hence everywhere for the canonical quasi-psh functions. Conversely, $w\in \mathrm{PSH}(X,\beta )$ has finite supremum, and

These maps are inverse. The bounded correspondence is theorem 11.41.1. Off $S$ the currents agree; locality of the extended products identifies their measures there. Both measures give the analytic set $S$ zero mass by theorem 13.14.3, even when $w$ has additional poles or infinite mass. This proves the measure identity on $X$ without subtracting infinite pointwise values.

For an envelope competitor $v\le 0$ with $v\preceq \varphi $, the corresponding $w=v-\varphi $ satisfies $w\le -\varphi $ off $S$. Hence

uniformly in $v$. On a fixed finite cover by inner and outer coordinate balls, add smooth quadratic corrections making $w$ locally psh. Submean at a point attaining $\sup _ Xw$, with uniform correction and volume-ratio bounds, gives $\sup _ Xw\le C_1\int _ Xw^+\, \mathrm dV+C_2\le K$. Thus $v\le \varphi +K$ canonically. The function $\varphi $ itself is a competitor, so the ordinary envelope construction gives a genuine $\theta $-psh function between $\varphi $ and $\varphi +K$. The upper bound survives upper regularization because $\varphi +K$ is upper semicontinuous. Ordinary equivalence gives $P_\theta [\psi ]=P_\theta [\varphi ]=\psi $. For $\varepsilon <1$ sufficiently close to one, $\beta -(1-\varepsilon )\theta >0$, so $\varphi $ is an $\varepsilon \theta $-psh competitor below $\psi $. Hence $P_{\varepsilon \theta }(\psi )$ is not identically $-\infty $, as required in (16.28.67).

If $u\sim \varphi \sim \psi $, choose $K_ u\ge 0$ with $\psi -K_ u\le u$. For each $t\ge 1$, the $\theta $-psh function $a_ t=\psi -tK_ u$ obeys

This proves relative envelope-class membership in definition 16.28.61, including the convention at $t=1$ and common poles. The other containment follows from its defining order $u\preceq \psi \sim \varphi $. $\square$

Proof. The correspondence lemma and theorem 15.28.4 for the Hermitian form $\beta $ prove the first assertion, including attainment. This assertion allows additional poles and excess or infinite mass. For smooth positive $f$, theorem 14.2.1 gives a smooth $w$ with $\beta +\mathrm{dd}^{\mathrm{c}}w>0$ and $(\beta +\mathrm{dd}^{\mathrm{c}}w)^ n=c_\beta (f)f\, \mathrm dV$. The potential $\varphi +w$ belongs to the ordinary type and hence to all three classes. The class containments and the first assertion give equality of their coefficients; bounded-data normalization identifies $b_\beta (f)=c_\beta (f)$. The exact measure is finite.

It remains to prove strictness. Let $v$ be the smooth solution for $\theta $, and put $\sigma =\theta +\mathrm{dd}^{\mathrm{c}}v$, $\eta =\beta +\mathrm{dd}^{\mathrm{c}}w$. The one-sided subsolution gives $c_\beta (f)\le b_\theta (f)=c_\theta (f)$. If equality held, then on $X\setminus S$ the smooth function $a=\varphi +w-v$ would satisfy

The form $Q$ is smooth and strictly positive, so it defines a real second-order operator $L$ with positive definite principal matrix and no zeroth-order term. Also $a\to -\infty $ near $S$, so its maximum is attained inside a component $U$ of $X\setminus S$. The strong maximum principle makes $a$ constant on $U$. To recall its local argument, put $z=\max _ Ua-a\ge 0$. If a positive region of $z$ had a boundary zero inside $U$, choose a ball in that region touching the zero set, with closure in a coordinate chart. On its outer half-annulus, the barrier $e^{-A|x-y|^2}-e^{-AR^2}$ has strictly positive $L$ for large $A$. A sufficiently small multiple lies below $z$ on the two boundary spheres. A negative interior minimum of the difference would have nonnegative $L$, whereas its equation gives strictly negative $L$. Thus the comparison holds throughout the annulus and gives a positive inward derivative at the touching zero, contradicting $\nabla z=0$. Connectedness propagates the zero set to all of $U$. The boundary of $U$ is nonempty and lies in $S$: otherwise $U$ would be both open and closed in connected $X$, although $S\ne \varnothing $. Constancy then contradicts $a\to -\infty $ at that boundary. This proves strictness; no semipositivity of the divisor's smooth curvature representative is required. $\square$

Proof. If such a component existed, retain a sufficiently small positive multiple $E$ of it. Choose a canonical logarithm $\ell _ E$ and a smooth closed representative $\gamma _ E$ with $[E]=\gamma _ E+\mathrm{dd}^{\mathrm{c}}\ell _ E$. The multiple can be chosen so that $\theta -\gamma _ E>0$. Divisorial removal gives a canonical $(\theta -\gamma _ E)$-psh representative of $u-\ell _ E$, whose finite supremum implies $u\preceq \ell _ E$. The preceding theorem would then give

a contradiction. $\square$

Proof. The Poincaré–Lelong formula gives $[D]=\kappa +\mathrm{dd}^{\mathrm{c}}(\ell _ E/\lambda )$. Dimensions and the mass-one projective factor give

Thus the zero residual solves each smooth positive equation and fixes its unique scalar. The theorem identifies all three restricted coefficients, and the correspondence gives $\mathrm{MA}^{\mathrm{ext}}_ H(\ell _ E)=H_ E^ N$. Under the factor hypothesis, pullbacks of bounded $h_0$-psh potentials with mass tending to zero have masses multiplied respectively by $C$ and $C(1-\lambda )^ k$. Bounded submersion pullback and the product expansion therefore give zero lower mass for both backgrounds. A closed positive $h_0$ would instead have every bounded mass equal to $V_0$ by Stokes, so both backgrounds are nonclosed. $\square$