15.16 Minimal singularities in an arbitrary big background5961B4E1812C

Proof. Let $\pi :Y\to X$ be a modification resolving the singularities of $\psi $. The pulled-back current has the form

where $D$ is an effective real divisor, $[D]$ is its current of integration, and $\beta $ is a smooth real $(1,1)$-form on $Y$. Off $D$ we have $\beta \geq a\pi ^*\omega $, and by continuity this inequality holds everywhere. In particular, $\beta $ is semipositive.

We check that $\beta $ is big. For a blowup with smooth center and exceptional divisor $E$, choose a metric on $\mathcal O(-E)$ whose curvature is positive in the directions tangent to the projective fibers. Along $E$, the pullback of a Hermitian metric is positive in the complementary directions. Consequently, its sum with a sufficiently small positive multiple of this curvature form is a Hermitian metric: positivity holds first along $E$, then on a neighborhood of $E$, and finally on the compact complement, where the pullback is already positive. Iterating over the blowups, we obtain an effective real exceptional divisor $F$, with logarithmic potential $\zeta $, and a smooth Hermitian form $\kappa $ such that

At each step of the iteration, we pull back the smooth positive form obtained so far, apply the construction for one blowup, and add the pullback of the previous divisor to the new positive multiple of $E$. Hence $\beta +a\mathrm{dd}^{\mathrm{c}}\zeta \geq a\kappa $, so $\beta $ is big. This argument only involves Hermitian metrics and never requires a closed positive form on $Y$.

Fix a smooth positive volume form $\mathrm{d}V_ Y$ on $Y$, and write $\pi ^*\mathrm{d}V=J\, \mathrm{d}V_ Y$. The function $J$ is smooth, nonnegative and bounded, and it is positive off the exceptional locus. Define

This function is nonzero and belongs to $L^ p(\mathrm{d}V_ Y)$. Indeed, $\psi $ is bounded above, and the change of variables formula gives

the analytic exceptional sets do not affect these integrals. Theorem 15.3.3, applied on $Y$ in the semipositive big background $\beta $, gives a bounded $s\in \mathrm{PSH}(Y,\beta )$ with

The function $\pi ^*\psi +s$ is $\pi ^*\theta $-plurisubharmonic, because the corresponding current is $\beta _ s+[D]\geq 0$, where $\beta _ s=\beta +\mathrm{dd}^{\mathrm{c}}s$. By theorem 9.11.5 it descends to some $u\in \mathrm{PSH}(X,\theta )$, and $u\sim \psi $ because $s$ is bounded. Off the exceptional set and the pole set of $\psi $, the change of variables formula gives the required equation. Neither the extended measure on the left nor the measure $\mathrm{e}^{\lambda u}f\, \mathrm{d}V$ on the right charges these analytic sets, so the equation holds on all of $X$. $\square$

Proof. Step 1: approximating singularity types. Apply theorem 5.36.5 to $V_\theta $. Set $\delta _ j=2^{-j}$ and pass to a subsequence whose curvature errors are at most $\delta _ j$. Denote its potentials by $\widetilde\psi _ j$; thus $\widetilde\psi _ j\downarrow V_\theta $ and $\theta _{\widetilde\psi _ j}\geq -\delta _ j\omega $. Set

By Hartogs' lemma, $\sup _ X\widetilde\psi _ j\to \sup _ XV_\theta =0$, so $a_ j\to 0$. A common resolution of the singularities of $\widetilde\psi _ j$ and $\rho $ shows that $\psi _ j$ has a smooth remainder after resolution. Moreover,

The upper inequality holds because $\psi _ j$ is $\theta $-plurisubharmonic and nonpositive. Subtracting $a_ j$ explicitly keeps the normalization separate from the fixed barrier $\rho $.

Step 2: uniform bounds. By lemma 15.16.1 there are $u_ j\sim \psi _ j$ satisfying

Each $u_ j$ is bounded below by $\rho $ minus a constant depending on $j$, so proposition 15.14.2 gives a uniform upper bound for $\sup _ Xu_ j$.

There is also a uniform lower bound. We apply lemma 15.14.1 with the barrier $\rho $ replaced by $\rho -1-\sup _ ja_ j$ and with the reference $\psi _ j-1$, which lies between this barrier and zero. The lemma gives $v_ j$ with $\psi _ j-1\leq v_ j\leq \psi _ j\leq 0$ and $\mathrm{MA}_\theta (v_ j)\geq mf\, \mathrm{d}V$, where $m>0$ is independent of $j$. Put $w_ j=v_ j+\lambda ^{-1}\log m$. Then

and the exponential order in corollary 15.13.3 gives $u_ j\geq w_ j$. Thus, for a uniform constant $C$,

After extraction, $u_ j$ converges in $L^1$ and almost everywhere to a $\theta $-plurisubharmonic function $u$, and the preceding bounds give $V_\theta -C\leq u\leq V_\theta +C$.

Step 3: the equation for the limit. We identify the equation using envelopes. Put $A_ j=\sup _{k\geq j}a_ k$ and, for $N\geq j$, define

Both functions lie above the $\theta $-plurisubharmonic function

which is a competitor for every $p_{j,N}$; in particular, no envelope is identically $-\infty $. All these functions are locally bounded on $\Omega $, uniformly in $j$ and $N$. Contact support (theorem 15.8.6), together with a partition of the contact set according to the first index of contact, gives

No factor $N$ appears here: on each piece the envelope equals a single $u_ k$, and ordered contact bounds its measure by the measure of that $u_ k$. Similarly, locality and ordered contact for a finite maximum give

Now let $N\to \infty $, and set $p_ j=\lim _ Np_{j,N}$ and $t_ j=(\lim _ Nt_{j,N})^*$. Continuity of local bounded products along monotone sequences passes both inequalities to these limits. Next, $p_ j$ increases to a regularized limit $p$, while $t_ j$ decreases to $u$ by Hartogs' lemma. The bound by $b_ j$ gives $p\sim V_\theta $, and $p\leq u$. A second application of the monotone continuity gives

These inequalities hold on $\Omega $, and they extend across $Z$ because no measure involved charges $Z$. Thus $\mathrm{MA}_\theta (p)\leq \mathrm{e}^{\lambda (p-u)}\mathrm{MA}_\theta (u)$, and the exponential order in corollary 15.13.3 gives $p\geq u$. Hence $p=u$, and the equation follows. Applying the same order in both directions proves uniqueness. Note that the Monge–Ampère operator was only ever applied along monotone sequences that are locally bounded on $\Omega $. $\square$

Proof. The normalized potential satisfies

Fix a Gauduchon metric $\gamma $ and write $\gamma ^ n=G\, \mathrm{d}V$. Since $v_\lambda \leq 0$ and $0<\lambda \leq 1$, (15.14.3) gives

The integrals on the left have a positive lower bound independent of $\lambda $. Otherwise, the compactness of normalized quasi-plurisubharmonic functions would give an almost everywhere convergent sequence of the normalized $v_\lambda $ whose integrals tend to zero, with a $\theta $-plurisubharmonic limit; since $f$ is nonzero, dominated convergence would give a positive limit integral. Thus $c_\lambda \leq C$.

To control the singularity type, we use uniform exponential integrability (lemma 5.26.2) and Hölder's inequality to choose $\varepsilon >0$ and $1<q<p$ such that $\| \mathrm{e}^{-\varepsilon v_\lambda }f\| _ q$ is uniformly bounded. Lemma 15.14.1, applied with the reference $V_\theta -1$ and the barrier $\rho -1$, gives $\eta _\lambda \in \mathrm{PSH}(X,\theta )$ with

for a fixed $m>0$. The scalar order in corollary 15.13.3, together with $\mathrm{MA}_\theta (v_\lambda )\leq c_\lambda f\, \mathrm{d}V$, gives $c_\lambda \geq m$. Set $D=\varepsilon ^{-1}\max \{ 0,\log (C/m)\} $ and $w_\lambda =\eta _\lambda -D$. Then

since $\lambda v_\lambda \leq 0$ and $\eta _\lambda \leq 0$. The exponential order in corollary 15.13.3 gives $v_\lambda \geq w_\lambda $, which proves the uniform bounds with $B=1+D$. $\square$

Proof. Apply lemma 15.16.5 with $\lambda =1/j$, and write $v_ j=v_{1/j}$ and $c_ j=c_{1/j}$. Extract a subsequence with $v_ j\to v$ in $L^1$ and almost everywhere, and with $c_ j\to c\in [m,C]$. The limit is normalized, $\theta $-plurisubharmonic, and has minimal singularities. The densities $c_ j\mathrm{e}^{v_ j/j}f$ have a common $L^ p$ bound. Hence corollary 15.15.3, applied with the reference $V_\theta $, upgrades the convergence to uniform convergence of differences on $\Omega $ and identifies the weak $L^ p$ limit of these densities with the density of $\mathrm{MA}_\theta (v)$.

At almost every point, $v_ j$ converges to a finite value, so $v_ j/j\to 0$. Since $v_ j\leq 0$ and $c_ j\leq C$, dominated convergence gives $c_ j\mathrm{e}^{v_ j/j}f\to cf$ in $L^ p$. Therefore $\mathrm{MA}_\theta (v)=cf\, \mathrm{d}V$. The uniqueness of the constant follows from corollary 15.13.3. $\square$