Theorem 16.19.12F8D75599A90

Proof. After reducing to one obstacle and passing to a modification, we bound the smoothing error $H_ u(x,r)-u(x)$ off the divisor, first linearly and then quadratically in $r$, as outlined above; the quadratic bound and (16.17.6) then bound $D_ hu$, and the Sobolev regularity follows.

Reduction to one obstacle with bounded complex Hessian. Choose a Kähler form $\kappa \geq \theta $, for instance a large multiple of $\omega $, and put $f=P_\kappa (\min _ jf_ j)$. By theorem 16.15.24, $f$ is $C^1$ and has bounded complex Hessian. Every $\theta $-psh function below $\min _ jf_ j$ is $\kappa $-psh, and hence lies below $f$. Conversely $f\leq \min _ jf_ j$. These two inclusions of competitor families give $P_\theta (f)=u$.

Passage to a modification. Put $Z=E_{\mathrm{nK}}(\{ \theta \} )$. By theorem 10.5.4, $Z$ is the pole set of a function $\phi _1\in \mathrm{PSH}(X,\theta )$ with analytic singularities such that $\theta +\mathrm{dd}^{\mathrm{c}}\phi _1\geq 2\delta _1\omega $ for some $\delta _1>0$. We apply lemma 6.17.1 to $\phi _1$, regarded as an element of $\mathrm{PSH}(X,\theta -2\delta _1\omega )$, with an integer $m_0$ so large that the constant $c'$ of that lemma satisfies $c'/m_0\leq \delta _1$, and we let $\phi _0$ be the resulting function. Then $\theta +\mathrm{dd}^{\mathrm{c}}\phi _0\geq \delta _1\omega $, and $\phi _0$ has gentle analytic singularities (definition 6.14.12) of type $(m_0^{-1},\mathfrak a)$, where $\mathfrak a=\mathcal I(m_0\phi _1)$. The pole set of $\phi _0$ is the zero set of $\mathfrak a$, and it equals $Z$. Indeed, near a point where $\phi _1$ is finite, $\phi _1$ is bounded, because it has analytic singularities, so $\mathfrak a$ is the unit ideal there; thus the pole set of $\phi _0$ is contained in $Z$. Conversely, $\theta +\mathrm{dd}^{\mathrm{c}}\phi _0$ is a Kähler current whose potential has analytic singularities, so its pole set contains $Z$ by the definition of the non-Kähler locus (definition 10.5.1).

Let $\pi \colon Y\to X$ be a log resolution of $\mathfrak a$, that is, a principalization of $\mathfrak a$ by finitely many blowups with smooth centers (theorem 9.6.7), and write $\pi ^*\mathfrak a=\mathcal O_ Y(-D_0)$; the support of the effective divisor $D_0$ is $\pi ^{-1}(Z)$. By definition 6.14.12, locally on $Y$ the function $\phi _0\circ \pi $ is $m_0^{-1}\log |s_{D_0}|^2$ plus a smooth function. By theorem 9.6.7, $\pi $ is an isomorphism over $X\setminus Z$, and all its centers lie over $Z$. If $\sigma $ blows up a compact Kähler manifold with Kähler form $\omega '$ along a closed submanifold of positive codimension, with exceptional divisor $E$, then $\sigma ^*\omega '+\varepsilon \eta $ is Kähler for every sufficiently small $\varepsilon >0$, where $\eta $ is the curvature of a suitable smooth quadratic Hermitian metric on $\mathcal O(-E)$ (proposition 6.14.11). Applying this to the successive blowups of $\pi $, with rational $\varepsilon $, and pulling the forms back to $Y$, we obtain a Kähler form

where $\eta _\ell $ is the curvature of a smooth metric on $\mathcal O(-E_\ell )$ for effective exceptional divisors $E_\ell $, and the coefficients $\varepsilon _\ell >0$ can be taken rational. The pulled-back current $\pi ^*(\theta +\mathrm{dd}^{\mathrm{c}}\phi _0)$ is at least $\delta _1\pi ^*\omega $. Let $s_{E_\ell }$ be the canonical section of $\mathcal O(E_\ell )$, with norm $|s_{E_\ell }|$ computed in the dual metric. Choose a rational number $0<\delta _2\leq \delta _1$ and add $\delta _2\sum _\ell \varepsilon _\ell \log |s_{E_\ell }|^2$ to the pulled-back potential $\phi _0\circ \pi $. Since, by the Poincaré–Lelong formula (theorem 5.5.6),

the resulting current is at least $\delta _2\omega _ Y$. Locally its potential is still of the form $\sum _ j\beta _ j\log |g_ j|^2$ plus a smooth function, where the $g_ j$ are local equations of the components of an effective divisor with support $\pi ^{-1}(Z)$ (the exceptional divisors lie over $Z$) and the $\beta _ j$ are positive rational numbers. Replacing $\omega _ Y$ by $\delta _2\omega _ Y$, we may assume that the lower bound is $\omega _ Y$ itself.

By proposition 10.3.9, pullback is a bijection from $\mathrm{PSH}(X,\theta )$ onto $\mathrm{PSH}(Y,\pi ^*\theta )$. Since $f$ is continuous, a potential lies below $f$ if and only if its pullback lies below $f\circ \pi $, so $u\circ \pi =P_{\pi ^*\theta }(f\circ \pi )$. The pullback $f\circ \pi $ is $C^1$ and has bounded complex Hessian. Indeed, off $\pi ^{-1}(Z)$ the map $\pi $ is a local biholomorphism, and since its differential is bounded, the chain rule bounds the coefficients of $\pi ^*(\kappa +\mathrm{dd}^{\mathrm{c}}f)$ there. Locally, $f\circ \pi $ plus a smooth local potential of $\pi ^*\kappa $ is a bounded plurisubharmonic function. Applying theorem 7.4.1 to it and to $\log |g|$, where $g$ is a local equation of the hypersurface $\pi ^{-1}(Z)$, shows that this logarithm is integrable for the trace measure of its complex Hessian, so the trace measure gives $\pi ^{-1}(Z)$ zero mass. Hence the positive current $\pi ^*\kappa +\mathrm{dd}^{\mathrm{c}}(f\circ \pi )$, whose coefficients are bounded off $\pi ^{-1}(Z)$, has bounded coefficients everywhere. We may therefore work on $Y$ with the pulled-back form and obstacle, and return through the isomorphism over $X\setminus Z$ at the end.

From now on we write $X$, $\theta $, $\omega $, $f$ and $u$ for $Y$, $\pi ^*\theta $, the Kähler form just constructed, $f\circ \pi $ and $u\circ \pi $, and again put $\alpha =2\pi \theta $ and $\widehat\omega =2\pi \omega $. Adding a constant to the obstacle, we may assume that $0\leq f\leq B$ with $B\geq 1$. Let $\phi $ be the potential $\phi _0\circ \pi +\delta _2\sum _\ell \varepsilon _\ell \log |s_{E_\ell }|^2$ constructed above, minus a constant chosen so that

here $\phi \leq u$ holds because $\phi $ is a competitor for the envelope. Cover $X$ by finitely many coordinate charts, with smaller charts still covering $X$, on each of which

where the $g_ j$ are local equations of the components of the divisor meeting the chart (there are none if the chart does not meet the divisor) and $\sigma $ is smooth. Write $D$ for the divisor and $U=X\setminus D$. In particular $\mathcal D\phi $ is uniformly bounded on $U$, although $\phi $ tends to $-\infty $ near $D$. All bounds below are uniform on this finite atlas; only the explicitly displayed functions of $\phi $ are allowed to grow.

A first-order smoothing bound. Let $H=H_ u$ and $\lambda =\lambda _ u$ be as in lemma 16.17.1. The initial joint estimate (16.17.3) improves after adding $K_1r$. Indeed, the vertical Levi form of $K_1|w|$ is $K_1/(4r)$, and Young's inequality absorbs the bounded mixed term at a horizontal cost $Cr$. Hence, for a fixed sufficiently large $K_1$ and small $r$,

as joint forms; the right side is pulled back from $X$.

Fix a small $\delta _0>0$; it will be decreased finitely many times, each time depending only on the fixed data. For $0<\delta <\delta _0$ and $c>0$ define on $U$

The infimum over $t$ is finite and attained, because $H\geq u$, $u$ is locally bounded on $U$, and $-c\log (t/\delta )\to +\infty $ as $t\to 0$. At a minimizing radius, including the upper end point, the logarithmic slope $\lambda +K_1t$ of the nondecreasing part $H+K_1t$ is at most $c$, so $\lambda \leq c$. We apply lemma 16.18.1 in the variable $s=\log t$, with the constant upper end point $\log \delta $, to $H(x,\mathrm e^ s)+K_1\mathrm e^ s$ plus a smooth local potential of $\alpha $; by (16.19.3), the Levi bound required in the lemma holds with $\Lambda =(Ac+C\delta )\widehat\omega $. Moreover, since $u\leq f$ and the kernel of $\Psi $ is positive, evaluation at $t=\delta $ gives $\psi _{c,\delta }\leq \Psi _ u(x,\delta )+K\delta ^2\leq \Psi _ f(x,\delta )+K\delta ^2$, and (16.17.5) bounds the right side by $f+C\delta ^2$. Thus

Put $e=Ac+C\delta $ and $\tau =e/(1+e)$. Then $(1-\tau )(\psi _{c,\delta }-C\delta ^2)+\tau \phi $ is $\theta $-psh on $U$, by (16.19.2), since $-(1-\tau )e+\tau =0$; and it lies below $(1-\tau )f\leq f$, because $\phi \leq 0\leq f$. It is bounded above near $D$, so by theorem 5.11.1, applied after adding smooth local potentials of $\alpha $, it extends across $D$; the extension is given by upper limits and therefore still lies below the continuous obstacle. By the maximality of the envelope, this extension lies below $u$. Rearranging, and using $\psi _{c,\delta }\leq B+C\delta ^2$ so that no unbounded factor is divided by $1-\tau $, we obtain

Now take $c=\delta $, so that $e$ and $\tau $ are $O(\delta )$, and let $t_0$ be a minimizing radius for $\psi _{\delta ,\delta }(x)$. The two nonnegative quantities $H(x,t_0)+K_1t_0-u(x)$ and $\delta \log (\delta /t_0)$ add up to $\psi _{\delta ,\delta }(x)-u(x)+K_1\delta $, so (16.19.4) gives, with $R=C(1+B+|\phi |)$ for a suitable uniform $C$,

Since $H$ is nondecreasing in the radius, the choice $\delta =r\mathrm e^{R(x)}$ yields

Since $\phi \leq -B-1$, we have $1+B\leq |\phi |$, so we may take $R(x)=B_0|\phi (x)|$ with a fixed large $B_0$.

This estimate also improves the bound for the trace mass. Enlarging $B_0$ if necessary, we may assume $B_0\beta _ j\geq 1$ for all $j$; since $|g_ j|<1/2$, the formula for $\phi $ then gives $\mathrm e^{-R(x)}\leq C|g_ j(x)|^2$ for each local equation $g_ j$. As the first derivatives of the $g_ j$ are bounded, there is a fixed small $S>0$ such that, for $|y-x|\leq S\mathrm e^{-R(x)}$, the ratios $|g_ j(y)|/|g_ j(x)|$ lie between $1/2$ and $3/2$. Hence $|\phi (y)-\phi (x)|\leq C$ there. On a chart that does not meet the divisor, $\phi $ is smooth, and the same oscillation bound holds after decreasing $S$. By (16.19.2), $|u(y)|\leq C(1+|\phi (x)|)$ on that ball. The comparison with radial convolution in lemma 16.17.1 therefore transfers (16.19.5) to the Euclidean radial mean of the local plurisubharmonic potential $p=u+q$, with an additional error $Cr^2(1+|\phi (x)|)$.

We also need the following consequence of the Jensen–Lelong formula, with positive constants $c_0$ and $0<c_1<1$ independent of $x$ and $r$:

Indeed, the spherical means of $p$ about $x$ are nondecreasing in the radius and tend to $p(x)$. Letting $r_1\to 0$ in (6.7.7) and using that $D_ p(x,\cdot )$ is nondecreasing, we see that the spherical mean of radius $\varrho $ exceeds $p(x)$ by at least $2\log (1/c_1)D_ p(x,c_1\varrho )$, which by (6.7.5) is a fixed positive multiple of $\varrho ^{2-d}\mu _ p(B(x,c_1\varrho ))$. It remains to integrate over the radii $\varrho =r\varrho '$ with $\varrho '$ in a fixed subinterval of $(0,1)$ on which $\chi (\varrho '^2)>0$, and to decrease $c_1$. We apply this inequality at a fixed multiple of $r$, decrease $S$, and use (16.19.5). This gives

Thus the improved joint estimate (16.17.4) is available for these radii.

Uniform joint estimates in divisor coordinates. On each chart put $G=(\prod _ jg_ j)^ m$, where an empty product equals one. By the formula for $R$ and (16.19.6), $M$ is bounded by a fixed product of negative powers of the $|g_ j|$, and differentiating $G$ loses only one power of one factor. Hence we can choose the integer $m$ so large that $M|G|$ and $M|\mathrm dG|$ are uniformly bounded. Next, by the same formula for $R$, we can choose an integer $k\geq 1$ so large that

with $T$ uniform on the finite cover. Put $\rho _\delta (x)=\delta \mathrm e^{-kR(x)}$, and take $\delta _0<S$, so that these radii lie in the range of (16.19.6). In the holomorphic coordinate $\xi $ defined by $w=G(x)\xi $, the range $0<|w|\leq \rho _\delta (x)$ corresponds to $|\xi |\leq T\delta $.

The differential of the change of variables $(x,\xi )\mapsto (x,G(x)\xi )$ sends $(v,\eta )$ to $(v,G\eta +\xi \, \mathrm dG(v))$. Substituting this in (16.17.4) and using the bounds just obtained, $|G|\leq 1$ and a uniform bound for $|\mathrm dG|$, we get

For example, the purely horizontal error is bounded by a constant times $M(|G|^2+|G||\mathrm dG|+|G||\mathrm dG|^2)|\xi |^2$, the mixed coefficient by $C|G||\xi |$, and the vertical coefficient by $C|G|^2|\xi |$. These estimates use only the boundedness of $M|G|$ and $M|\mathrm dG|$.

Now add $K_2|G\xi |^2$. Its Levi form is the square $K_2|G\eta +\xi \, \mathrm dG(v)|^2$, and the inequality $|a+b|^2\geq \tfrac 12|a|^2-|b|^2$ shows that it is at least $\frac{K_2}2|G|^2|\eta |^2$ minus a horizontal term $K_2|\mathrm dG|^2|\xi |^2|v|^2$. For $\delta _0$ small and $K_2$ fixed and large, this absorbs the last term of (16.19.7), and Young's inequality absorbs its mixed term at a cost $C|\xi |^2|v|^2$. Since $|\xi |\leq T\delta $, we have proved the uniform joint bound

All constants here are independent of the chart and of the point of $U$.

A second-order smoothing bound. For $c>0$ set

In the variable $s=\log |\xi |$, so that $t=|G|\mathrm e^ s$, the upper end point of the interval is $b(x)=\log \delta -kR(x)-\log |G(x)|$. The last summand is pluriharmonic on $U$, and $\mathcal D R=-B_0\mathcal D\phi $ is uniformly bounded there. Thus $|\mathcal D b|\leq C_ b\widehat\omega $ with a uniform constant $C_ b$. The logarithmic slope of the nondecreasing part $H+K_2t^2$ is $\lambda +2K_2t^2$, so at a minimizing radius it is at most $c$. Since $-c\log t=-cs-c\log |G|$ and $\log |G|$ is pluriharmonic on $U$, lemma 16.18.1, applied to (16.19.8) (again after adding a smooth local potential of $\alpha $), proves

Let $a=ckB_0$ and $C_3=C_ f+K+K_2$, with $C_ f$ from (16.17.5). Since $-c\log \rho _\delta =-c\log \delta -a\phi $, evaluation at $t=\rho _\delta (x)\leq \delta $, together with $\Psi _ u\leq \Psi _ f$ and (16.17.5), gives

the second inequality follows from (16.19.9) and $a(\alpha +\mathcal D\phi )\geq 0$, a consequence of (16.19.2). Put

Since $\nu (1+a)=(1-\nu )C_0$, we have $(1+a)\alpha +\mathcal DW\geq 0$, that is, $W$ is $(1+a)\theta $-psh. It lies below $f$, because $Q\leq f$, $\phi \leq 0$ and $f\geq 0$. As before, $W$ is bounded above near $D$, so it extends across $D$ by theorem 5.11.1, applied after adding smooth local potentials of $(1+a)\alpha $, and the extension, given by upper limits, still lies below the continuous function $f$. Consequently $W/(1+a)$, which lies below $f/(1+a)\leq f$, is a competitor for $u$, and $(1+a)u\geq W$. Since $u\leq B$ and $Q\leq B$, we can rearrange this without losing an unbounded positive term:

Now take $c=\delta ^2$, so that $a$ and $\nu $ are $O(\delta ^2)$, and let $t_0$ be a minimizing radius for $\psi (x)$. Writing $\psi (x)$ as the value at $t_0$ of the minimized expression and substituting it into (16.19.10), we obtain

Indeed, after rearrangement the left side consists of the two nonnegative terms $H(x,t_0)+K_2t_0^2-u(x)$ and $-\delta ^2\log (t_0/\delta )$ (recall $H\geq u$ and $t_0\leq \delta $), while every term on the right is bounded by $C\delta ^2(1+B+|\phi |)$.

Fix a compact set $F\Subset U$ and a number $M_2\geq \sup _ FR_2$. For every sufficiently small $r$, choose $\delta =r\mathrm e^{M_2}$. Then $r\leq t_0$ at every $x\in F$, and since $H$ is nondecreasing in the radius we obtain the uniform quadratic estimate

Regularity. Divide (16.19.12) by $r^2$: on $F$ we have $0\leq (\Psi _ u(x,r)-u(x))/r^2+K\leq M_2\mathrm e^{2M_2}$. Pairing with nonnegative test functions supported in the interior of $F$ and letting $r\to 0$, (16.17.6) shows that, in the sense of distributions on the interior of $F$, $a_\chi D_ hu+K$ lies between zero and the constant $M_2\mathrm e^{2M_2}$. In particular $|\langle a_\chi D_ hu,\varphi \rangle |\leq (K+M_2\mathrm e^{2M_2})\int \varphi \, \mathrm dx$ for these test functions $\varphi \geq 0$. Since every point of $U$ lies in the interior of such a set $F$, corollary 2.8.2(2), applied in the charts with Lebesgue measure, shows that $D_ hu$ is an $L^\infty _{\mathrm{loc}}$ function on $U$. The positive current $\alpha +\mathcal Du$ therefore has locally bounded $h$-trace, so in a fixed chart each of its diagonal coefficient measures is bounded by a constant times Lebesgue measure. The Cauchy–Schwarz inequality for positive Hermitian matrices bounds the off-diagonal coefficient measures as well. Thus all complex coefficients of $\alpha +\mathcal Du$, and hence those of $\mathcal Du$, are locally bounded on $U$.

It remains to obtain the real Sobolev regularity; note that no weak first derivatives of $u$ are known in advance. Let $V\Subset U$ be a coordinate ball. The complex second derivatives $u_{i\bar j}$ are bounded on $V$, so the Euclidean Laplacian $\Delta u=4\sum _ iu_{i\bar i}$ is a bounded function there. Let $V'$ be a smaller concentric ball with $\overline{V'}\subset V$, choose $\vartheta \in C^\infty _ c(V)$ equal to one near $\overline{V'}$, and let $N=E_{2n}*g$ be the fundamental potential of the bounded compactly supported function $g=\vartheta \Delta u$, where $E_{2n}$ is the fundamental solution of proposition 3.1.13. By proposition 3.4.11, applied to the positive and negative parts of $g\, \mathrm dx$, we have $\Delta N=g$ in the sense of distributions. Hence $u-N$ is distributionally harmonic near $\overline{V'}$, and by Weyl's lemma (theorem 3.1.11) it is smooth there. It therefore suffices to show that $N\in W^{2,p}(V')$ for $1<p<\infty $.

The Hessian estimate of theorem 2.9.12 concerns compactly supported smooth functions, so we mollify. Let $g_\varepsilon $ be the mollifications of $g$, and put $N_\varepsilon =E_{2n}*g_\varepsilon $. For small $\varepsilon $ the $g_\varepsilon $ are smooth and supported in a fixed compact set, and $g_\varepsilon \to g$ in $L^ s$ for every $1\leq s<\infty $, by lemma 2.5.1. By proposition 3.1.13, $N_\varepsilon $ is smooth and $\Delta N_\varepsilon =g_\varepsilon $. Moreover $\partial _ a N_\varepsilon =E_{2n}*\partial _ ag_\varepsilon =(\partial _ aE_{2n})*g_\varepsilon $, where the second identity follows by integration by parts outside a small ball about the singularity of $E_{2n}$, whose boundary term tends to zero. Fix $q_0$ with $1<q_0<2n/(2n-1)$, and let $q_0'=q_0/(q_0-1)$. The functions $E_{2n}$ and $\partial _ aE_{2n}$, the latter bounded by $C|y|^{1-2n}$, belong to $L^{q_0}$ on bounded sets, so Hölder's inequality gives

Thus $N_\varepsilon \to N$ and $\partial _ aN_\varepsilon \to (\partial _ aE_{2n})*g$ uniformly on $V$. Passing to the limit in the sense of distributions, we see that $N$ is bounded on $V$ and that its first distributional derivatives there are the bounded functions $(\partial _ aE_{2n})*g$. Choose $\vartheta _1\in C^\infty _ c(V)$ equal to one on $V'$. For small $\varepsilon ,\varepsilon '$, the function $\vartheta _1(N_\varepsilon -N_{\varepsilon '})$ is smooth with compact support, and (2.9.13) gives

By the convergences just proved, the right side tends to zero as $\varepsilon ,\varepsilon '\to 0$. Hence, along a sequence $\varepsilon _ k\to 0$, the second derivatives $\partial _ a\partial _ b(\vartheta _1N_{\varepsilon _ k})$ form a Cauchy sequence in $L^ p(\mathbb R^{2n})$ and converge in $L^ p$. The limits are supported in the fixed compact set $\operatorname{Supp}\vartheta _1$, so by Hölder's inequality the convergence also holds in the sense of distributions. On the other hand $\vartheta _1N_{\varepsilon _ k}\to \vartheta _1N$ uniformly, so $\partial _ a\partial _ b(\vartheta _1N_{\varepsilon _ k})\to \partial _ a\partial _ b(\vartheta _1N)$ in the sense of distributions. Therefore $\partial _ a\partial _ b(\vartheta _1N)$ is the $L^ p$ limit and belongs to $L^ p$, and since $\vartheta _1=1$ on $V'$, the second distributional derivatives of $N$ belong to $L^ p(V')$.

Hence $N\in W^{2,p}(V')$, and therefore $u\in W^{2,p}_{\mathrm{loc}}(U)$ for every $1<p<\infty $; the case $p=1$ follows on relatively compact sets because their volume is finite. Morrey's inequality (lemma 2.9.16), with $p>2n/(1-\gamma )$, gives a representative in $C^{1,\gamma }_{\mathrm{loc}}$ for every $\gamma <1$. This continuous representative coincides with $u$: by lemma 6.14.4(1), the quasi-plurisubharmonic function $u$ is at every point the limit of its averages over small coordinate balls centered there, and these averages converge to the continuous representative. Transferring these statements through the biholomorphism $Y\setminus \pi ^{-1}(Z)\to X\setminus Z$, under which all smooth metrics are locally equivalent and local Sobolev spaces are preserved (lemma 2.9.14(2)), proves the theorem. $\square$