5.27 Local products and capacity5E507B6993DC

Proof. First let $r=n$, and write $\mu _ j=\bigwedge _ i\mathrm{dd}^{\mathrm{c}}b_{i,j}$ and $\mu =\bigwedge _ i\mathrm{dd}^{\mathrm{c}}b_ i$. Work over a relatively compact ball on which $-M\leq a_ j,a\leq M$. On the product of this ball with a disk of radius $R$ in an auxiliary variable $w$, where $\log R^2>M+1$, set

These are locally bounded plurisubharmonic functions, also at $w=0$, and $v_ j\downarrow v$. Pulling back the functions $b_{i,j}$ and $b_ i$ to the product and applying theorem 5.27.1 in dimension $n+1$, we obtain

Since $\mu $ is closed, $T=\mathrm{dd}^{\mathrm{c}}(v\mu )$. As $\mu $ has top degree in the base variables, only derivatives in $w$ contribute. Testing against product test functions and applying Fubini's theorem therefore identifies $T$ with $\mu (\mathrm dz)$ times the circle measure

Here $\sigma _ t$ is the uniform probability measure on the circle $|w|=t$: the radial derivative jumps from zero to $2/t$ there, so the Euclidean Laplacian has total mass $4\pi $, and our normalization of $\mathrm{dd}^{\mathrm{c}}$ divides that mass by $4\pi $. This distributional computation is valid for every bounded Borel function $a$; no continuity in $z$ is needed. The same description holds for $T_ j$.

Choose a smooth radial compactly supported function $\vartheta (w)$, supported away from zero and from the circle $|w|=R$, that equals $\log |w|^2$ on all circles with $-M\leq \log |w|^2\leq M$. For every smooth compactly supported test function $\xi $ on the base,

The convergence $T_ j\to T$ therefore proves the top-degree assertion.

For $r<n$, wedge with $n-r$ constant positive $(1,1)$-forms, each the Hessian of a positive quadratic function, and apply the top-degree case. By polarization, the products of positive constant $(1,1)$-forms span the real constant forms of bidegree $(n-r,n-r)$, and multiplying them by smooth test functions recovers every coefficient of the currents. For quasi-plurisubharmonic functions, add the fixed smooth corrections to the decreasing sequences, apply the result to the corrected functions, and expand; the term involving the correction to $a_ j$ converges by the ordinary decreasing continuity of the products. Finally, the local Chern–Levine–Nirenberg bounds show that these finitely many products have uniformly bounded variation on compact sets. This allows smooth forms converging locally in $C^\infty $ to replace fixed forms in the tests. $\square$

Proof. For $n\geq 2$, we use Cegrell's decomposition theorem. On a bounded hyperconvex domain in $\mathbb C^ n$, $n\geq 2$, let $\mathcal E_0$ consist of the bounded negative plurisubharmonic functions with zero boundary values and finite total Monge–Ampère mass. The theorem states that every positive compactly supported measure $\tau $ has a decomposition

where $\tau _ s$ is a positive measure carried by a pluripolar set. If $\tau $ is non-pluripolar, then $\tau _ s=0$, so $\tau \ll (\mathrm{dd}^{\mathrm{c}}v)^ n$.

In dimension one, the assertion follows directly from the Green potential. Let $a\leq 0$ be the Green potential of $\tau $ on the disk, normalized by $\mathrm{dd}^{\mathrm{c}}a=\tau $. Then $v=\mathrm{e}^ a$ is bounded and subharmonic, and

To justify this inequality, set $a_ M=\max (a,-M)$. The chain rule (5.27.9) in dimension one gives $\mathrm{dd}^{\mathrm{c}}\mathrm{e}^{a_ M}\geq \mathrm{e}^{a_ M}\mathrm{dd}^{\mathrm{c}}a_ M$, and the right-hand side is at least $\mathrm{e}^ a\tau $ on $\{ a>-M\} $. Let $M\to \infty $, using the decreasing convergence of the bounded subharmonic functions $\mathrm{e}^{a_ M}$ on the left and monotone convergence on the right. Since $\{ a=-\infty \} $ is polar and $\tau $ charges no polar set, $\mathrm{e}^ a>0$ almost everywhere for $\tau $. Thus $\tau \ll \mathrm{dd}^{\mathrm{c}}v$. $\square$

Proof. Put $f=u-v$ and write the Jordan decomposition on $B$ as $\mathrm{dd}^{\mathrm{c}}f=\alpha ^+-\alpha ^-$. Let $g^\pm \leq 0$ be the Green potentials of $\alpha ^\pm $ on $B$ with zero boundary values. The $L^1(B)$ norm of the Green kernel of the disk is bounded uniformly in its pole, so

Indeed, on the unit disk the kernel is $\log |(z-\zeta )/(1-\overline\zeta z)|^2$; its logarithmic singularity is uniformly integrable, and $|1-\overline\zeta z|\leq 2$. Scaling gives the bound on $B$.

The distribution $h=f-g^++g^-$ is harmonic, and the interior mean-value estimate gives

The identity $u+g^-=v+g^++h$ holds almost everywhere, hence everywhere by lemma 5.24.3. Thus $|u-v|\leq -g^+-g^-+|h|$ off the polar sets of $u$ and $v$. Let $0\leq a\leq 1$ be a subharmonic test function on $B$. Since the measure $\mathrm{dd}^{\mathrm{c}}a$ charges no polar set, this pointwise bound, the weighted Chern–Levine–Nirenberg estimate and (5.27.3) yield

Restricting the integral to $B'\cap \{ |u-v|>t\} $, dividing by $t$ and taking the supremum over $a$ proves (5.27.12). $\square$

Proof. It suffices to prove $\Delta f(K)\geq 0$ for every compact polar set $K\subset B$; restriction to smaller disks and inner regularity then give the assertion. Choose open neighborhoods $O_ j$ of $K$ with smooth boundaries such that $\overline O_{j+1}\subset O_ j\Subset B$ and $\bigcap _ j\overline O_ j=K$. Let $h_ j$ be the function that equals one on $\overline O_ j$, is harmonic on $B\setminus \overline O_ j$, and vanishes on $\partial B$. The Dirichlet problem on this smooth domain shows that $h_ j$ is continuous with $0\leq h_ j\leq 1$. Its outward normal derivative on $\partial O_ j$ is nonpositive, so the jump of the derivative across $\partial O_ j$ gives $\Delta h_ j\leq 0$ on $B$ in the sense of distributions. The maximum principle gives $h_{j+1}\leq h_ j$.

Since $0\leq h_ j\leq 1$, the sequence converges in $L^1_{\mathrm{loc}}(B)$. Let $h$ be the superharmonic representative of this limit; it may differ from the pointwise limit on $K$. It is harmonic off $K$, so its Laplacian is supported on the polar set $K$. On the other hand, $-h$ is bounded and subharmonic, so its Laplacian charges no polar set, by theorem 5.27.1. Hence $h$ is harmonic on $B$. Since $0\leq h\leq h_1$ and $h_1$ tends to zero at $\partial B$, the maximum principle gives $h=0$. It follows that $h_ j\to 0$ smoothly on compact subsets of $B\setminus K$, while $h_ j=1$ on $K$.

Choose $0\leq \chi \in C_ c^\infty (B)$ equal to one near $K$. For large $j$, regularize $h_ j$ by convolution on a neighborhood of $\operatorname {supp}\chi $, obtaining smooth functions $h_{j,\epsilon }$ with $\Delta h_{j,\epsilon }\leq 0$. Since $f\leq 0$,

For fixed $j$, the convolutions converge uniformly to $h_ j$ on $\operatorname {supp}\chi $, because $h_ j$ is continuous. On the support of the derivatives of $\chi $, the function $h_ j$ is harmonic for all large $j$. We may therefore first let $\epsilon \downarrow 0$ and then let $j\to \infty $. The smooth convergence away from $K$ makes the right-hand side tend to zero, and dominated convergence for the signed measure $\Delta f$ makes the left-hand side tend to $\Delta f(K)$. This proves $\Delta f(K)\geq 0$. $\square$

Proof. Apply lemma 5.27.14 to $u-v-C$ in local coordinates. Since smooth functions have zero polar part, linearity of the polar part gives the first assertion. Applying it to $u-\max (u,v)$ and to $v-\max (u,v)$ gives the inequality $\leq $ in (5.27.16).

For the reverse inequality, add a common smooth correction to make $u$ and $v$ subharmonic on a coordinate disk. On a smaller disk, let $g$ be the Green potential of the restriction of $\lambda =\mu _ u\wedge \mu _ v$ to that disk, so that $\mathrm{dd}^{\mathrm{c}}g=\lambda $. The distributions $u-g$ and $v-g$ have positive Hessians and hence subharmonic representatives $a$ and $b$. Since $\max (u,v)=g+\max (a,b)$ almost everywhere, the Hessian of $\max (u,v)$ dominates $\lambda $. Restricting this inequality to a polar Borel set carrying $\lambda $ proves the reverse inequality on the smaller disk, and these local statements give it on the whole surface. $\square$

Proof. Apply the classical theorem on products with unbounded potentials, together with its monotone continuity statement, in [Chapter III, Theorem 4.5 and Proposition 4.9, pp. 152–156, DemCADG], with the underlying current $T=1$. For every choice of $m$ of the functions, that theorem requires the intersection of their unbounded loci to have zero Hausdorff measure of dimension $2n-2m+1$. This intersection lies in $S$, and $2n-2m+1\geq 1$, so the measure is zero. The same hypothesis gives local integrability of the weighted current. Symmetry, multilinearity and locality on open sets follow from the local construction of the currents and decreasing regularization. $\square$

Proof. Fix a compact set $E\subset S$ and $\varepsilon >0$, and put $w_ c=\max \{ b-c,(1+\varepsilon )a\} $ for $c>0$. Since $a=-\infty $ on the compact set $E$, upper semicontinuity gives a neighborhood of $E$ on which $a<-(C+c)/\varepsilon $ and $a\leq b+C$. Off $S$, on this neighborhood, $b-c-(1+\varepsilon )a\geq -C-c-\varepsilon a>0$, while on $S$ the second branch of the maximum equals $-\infty $. Thus $w_ c=b-c$ near $E$, and locality on open sets gives $(\mathrm{dd}^{\mathrm{c}}w_ c)^ n_{\mathrm{full}}(E)=(\mathrm{dd}^{\mathrm{c}}b)^ n_{\mathrm{full}}(E)$.

As $c\to \infty $, the functions $w_ c$ decrease to $(1+\varepsilon )a$. Theorem 5.27.18 applies to every $w_ c$ with the same set $S$, so their full measures converge weakly to $(1+\varepsilon )^ n(\mathrm{dd}^{\mathrm{c}}a)^ n_{\mathrm{full}}$. Since the mass of a compact set is upper semicontinuous under weak convergence of positive measures, we obtain

Here we apply the semicontinuity after multiplying the measures by a fixed nonnegative cutoff function with compact support, equal to one near $E$; both masses on $E$ are finite by the theorem. Letting $\varepsilon \downarrow 0$ and applying inner regularity to all compact sets $E\subset S$ gives the asserted inequality of measures. If $a-b$ is bounded near $S$, then also $\{ b=-\infty \} =S$, so the roles of $a$ and $b$ can be interchanged, and equality follows. $\square$

Proof. Shrink $U$ so that $|h_ j|\leq 1$ and the unit $g$, its inverse and the first derivatives of $g$ and of all $h_ j$ are bounded. For each $j$, $|f|^{1/m_0}\leq C|h_ j|^{m_ j/m_0}\leq C|h_ j|$. Logarithmic differentiation of the factorization therefore gives

which proves the first bound.

For small $\zeta \in \mathbb C$, let $\mu _\zeta $ be the Euclidean volume measure of the fiber divisor $\{ f=\zeta \} $, counted with multiplicity. As $\zeta \to 0$, we have $\log |f-\zeta |\to \log |f|$ in $L^1_{\mathrm{loc}}$: by the local compactness theorem recalled at the beginning of section 5.25, it suffices to identify subsequential limits, and the convergence is locally uniform off $Z$, which identifies every such limit and excludes convergence to $-\infty $. The Poincaré–Lelong formula therefore gives weak convergence of the fiber currents, and hence of their Euclidean trace measures. Pairing with a fixed nonnegative cutoff function equal to one near $K$ shows that $\sup _{|\zeta |<r_ K}\mu _\zeta (K)<\infty $ for some $r_ K>0$.

The critical locus of $f$ has finitely many local components, and $f$ is constant on each of them. After shrinking $U$, every remaining critical component passes through $0$, where $f$ vanishes. Thus the small nonzero fibers are smooth. On $\{ |f|=r\} $, the coarea formula for the map $f$ to the circle of radius $r$ gives

The coarea formula for the function $|f|$, together with $|\nabla |f||=|\partial f|$, proves the last inequality. The same formulas hold in dimension one, where the fiber volume is the counting measure. $\square$

Proof. At a smooth point of a component of multiplicity $m_0$, outside the other components, multiplying the defining function of the component by a holomorphic $m_0$-th root of the remaining unit gives a normal coordinate $w_1$ with $f=w_1^{m_0}$. Coordinate changes with bounded derivatives and Jacobians preserve local first-order Sobolev integrability. With $r=|w_1|$ and $s=m_0\log r$, polar integration of the classical gradient gives, up to fixed positive factors,

Such smooth points occur in every neighborhood of $0$. This proves necessity, since changing the finite upper endpoint does not affect convergence.

For sufficiency, put

For small $\varepsilon >0$, these are $C^2$ plurisubharmonic functions decreasing to $v$. By convexity, $|\chi (s)|\leq C(1+|s|)$ for $s\leq A$, so the local exponential integrability of $\log |f|$, recalled in [SGPT], gives convergence in every local $L^ p$ space with $p<\infty $. By the chain rule,

Apply lemma 5.27.21 with the coarea formula, keeping two powers of $|\partial f|$ for the level estimate and bounding the other $p-2$ powers by the first inequality of the lemma. The local integral of $|\nabla v_\varepsilon |^ p$ over $K\cap \{ 0<|f|<r_ K\} $ is then bounded by

With $s=\tfrac 12\log (r^2+\varepsilon ^2)$, we have $\mathrm ds=r\, \mathrm dr/(r^2+\varepsilon ^2)$ and $r\leq (r^2+\varepsilon ^2)^{1/2}$, so this is at most

The integral over the part where $|f|\geq r_ K$ is uniformly bounded as well. Weak compactness in Sobolev spaces and the local $L^ p$ convergence therefore show that $v\in W^{1,p}_{\mathrm{loc}}$. Its weak gradient agrees with the classical one off $Z$, and $Z$ has Lebesgue measure zero, which proves the final assertion. $\square$

Proof. The limit $\ell \geq 0$ exists because $\chi '$ is nonnegative and nondecreasing. Near a smooth point of a component of multiplicity $m_0$, outside the other components, the coordinate reduction to $f=w_1^{m_0}$ makes the positive complex Hessian of $v$ off $Z$ a single factor in the normal direction. Its Euclidean traces in the two coordinate systems are uniformly comparable, so polar integration gives necessity. In general, $G=\chi ''(\log |f|)|\partial f|^2/|f|^2$ off $Z$. The coarea formula and lemma 5.27.21, now bounding the remaining $2p-2$ powers of $|\partial f|$ by the first inequality of the lemma, give

The substitution $s=\log r$ proves sufficiency. For $p=1$ the integral is always finite, since $\int _{-\infty }^ A\chi ''=\chi '(A)-\ell $.

For the distributional identity, use $s_\varepsilon $ and $v_\varepsilon $ from the preceding proof. Direct differentiation gives

Off $Z$, the first term tends locally uniformly to $G$. By the level estimate of lemma 5.27.21 and the same substitution, its integral over $K\cap \{ |f|<r_0\} $ is at most

This bound tends to zero if we first let $\varepsilon \to 0$ and then $r_0\to 0$, so the first term converges to $G$ in $L^1_{\mathrm{loc}}$.

The positive measures $\Delta s_\varepsilon $ converge weakly to $\Delta \log |f|$, with locally bounded masses, and the Poincaré–Lelong formula shows that this limit is a measure supported on $Z$. On $\{ |f|<r_0\} $,

while $\Delta s_\varepsilon \to 0$ uniformly on compact subsets of $\{ |f|\geq r_0\} $. Taking the same successive limits shows that $(\chi '(s_\varepsilon )-\ell )\Delta s_\varepsilon \to 0$ weakly. Since $v_\varepsilon \to v$ in $L^1_{\mathrm{loc}}$, passing to the limit in the formula for $\Delta v_\varepsilon $ gives the identity. If $\ell >0$, the measure $\ell \, \Delta \log |f|$ is nonzero and singular with respect to Lebesgue measure; uniqueness of the Lebesgue decomposition therefore proves the final criterion. $\square$

Proof. Here $\chi (s)=-(-s)^ a$ is of class $C^2$ on $(-\infty ,-1]$, so the preceding propositions apply with $A=-1$. Moreover, $\chi '(s)=a(-s)^{a-1}$ and $\chi ''(s)=a(1-a)(-s)^{a-2}$, so $\ell =0$. The two propositions give the assertion about the Laplacian and the first-order thresholds. At $a=1/2$, the first-order integral diverges logarithmically; for $p>2$, its exponential factor forces divergence.

For second derivatives, take a smooth point of a component of multiplicity $m$, outside the other components, with a normal coordinate $w_1$ such that $f=w_1^ m$. Write $F(r)=\chi (m\log r)$ with $r=|w_1|$. The real second derivative of $v_ a$ in the unit direction tangent to the circles $|w_1|=r$ equals $F'(r)/r=m\chi '(m\log r)/r^2$. Thus the integral of its absolute value over the normal disk is bounded below, up to a fixed positive factor, by

Hence the classical Hessian of $v_ a$ off the divisor is not locally integrable, whereas local integrability is necessary for $W^{2,1}_{\mathrm{loc}}$. Since smooth coordinate changes preserve this Sobolev class, the argument is complete. $\square$

Proof. Write $t=\log |z|^2$ and approximate $\max \{ t,\log r^2\} $ uniformly by smooth convex increasing functions $F_\epsilon (t)$ whose derivatives equal zero below $\log r^2-\epsilon $ and one above $\log r^2+\epsilon $. Their compositions with $t$ extend smoothly across zero. Since $\mathrm{dd}^{\mathrm{c}}t$ has rank $n-1$ off zero, the chain rule gives

The integral of $\mathrm d^{\mathrm c}t\wedge (\mathrm{dd}^{\mathrm{c}}t)^{n-1}$ over each sphere is one. Indeed, $\mathrm d^{\mathrm c}t$ integrates to one on the Hopf circle, and the induced form on $\mathbb P^{n-1}$ is $\mathrm{dd}^{\mathrm{c}}\log (1+|w|^2)$; its top power has density $(n-1)!\pi ^{-(n-1)}(1+|w|^2)^{-n}$ on the affine chart, whose integral is one. Thus each displayed radial measure has total mass $\int ( (F_\epsilon ')^ n)'\, \mathrm dt=1$. Its support shrinks to the sphere, and its rotation invariance identifies the limit as $\sigma _ r$. Uniform bounded-potential convergence identifies that limit with the product in the statement. Non-pluripolarity follows from theorem 5.27.1. $\square$

Proof. Fix concentric balls $B'\Subset B\Subset B_1\Subset \Omega $, and put $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$. If $\psi $ is a nonnegative smooth cutoff function supported in $B_1$ and equal to one near $\overline B$, integration by parts gives

Choose $M>0$ with $|u_ j|\leq M$ on $B$ for all $j$, and apply the first estimate in (5.27.3) to $u_ j+M$, whose values lie in $[0,2M]$. It bounds the top-degree mass of $B'$ by $C(2M)^{n-1}$ times the trace integral over $B$, which tends to zero; for $n=1$ the trace estimate itself suffices. Covering $K$ by finitely many such balls proves the lemma. $\square$

Proof. By convexity, $0\leq \chi _\eta '\leq 1$ and $\chi _\eta ''\geq 0$. The function $\chi _\eta $ vanishes on $(-\infty ,-\eta ]$ and equals $t$ on $[\eta ,\infty )$. Since $T\to +\infty $ as $\zeta \to 0$, we have $h=v-A$ near each point with $\zeta =0$, which gives the smooth extension. Off the hyperplane $\{ \zeta =0\} $, the function $q$ is pluriharmonic and

The second summand has rank at most one, and the first has rank at most $n-1$. Hence

The last factor has top degree in the variables $z'$, so the derivatives of $v$ in $\mathrm dT\wedge \mathrm d^{\mathrm c}T$ contribute nothing. Write $s=\log (|\zeta |^2/R^2)$. In polar coordinates, $\mathrm dq=\varepsilon \, \mathrm ds$ and $\mathrm d^{\mathrm c}q=\varepsilon \, \mathrm d\vartheta /(2\pi )$. At fixed $z'$, the change of variables from $s$ to $t=v-A-\varepsilon s$ gives (5.27.30). There is no contribution near $\zeta =0$, where $h=v-A$ and the form $\mathrm{dd}^{\mathrm{c}}h$ has rank at most $n-1$. Finally, $w_\eta $ is the derivative of $(\chi _\eta ')^ n$, whose values at $-\eta $ and $\eta $ are zero and one. $\square$

Proof. Translate $a$ to the origin and write $z=(\zeta ,z')\in \mathbb C\times \mathbb C^ m$, $m=n-1$. It suffices to construct measures of mass one and then multiply the potentials by $c^{1/n}$. For $\delta >0$, put $H_\delta (z')=\log (|z'|^2+\delta ^2)$. With $s=|z'|^2$, the complex Hessian of $H_\delta $ has the eigenvalue $(s+\delta ^2)^{-1}$ with multiplicity $m-1$ in the directions orthogonal to $z'$ and the eigenvalue $\delta ^2(s+\delta ^2)^{-2}$ in the radial direction. Taking the determinant and integrating in polar coordinates gives

where $\lambda _{2m}$ is Lebesgue measure. These probability measures tend to $\delta _0$ as $\delta \downarrow 0$, since their mass outside each ball about the origin tends to zero.

Choose $R>\max (1,\sup _\Omega |\zeta |)$ and a ball $P\subset \mathbb C^ m$ containing the projection of $\Omega $ to the $z'$ variables. Set

Apply lemma 5.27.29 on $\mathbb C\times \mathbb C^ m$ to these data, and let $u_ j=h_ j-\eta _ j$, where $h_ j$ is the resulting function. The convolution satisfies $0\leq \chi _{\eta _ j}(t)-\max (t,0)\leq \eta _ j$, and $q+\max (T,0)=\max (q,v_ j-A_ j)$. Both branches of this maximum are negative on $\Omega $, so

Each $u_ j$ is smooth and bounded on $\Omega $, since the second branch $v_ j-A_ j$ is bounded below on the bounded set $P$. The logarithm has finite $L^ p(\Omega )$ norm for every finite $p$, by polar integration on a fixed polydisk containing $\Omega $. Hence $u_ j\to 0$ in all these norms.

Let $B'\Subset B\Subset \Omega $ be concentric balls, and let $w\in \mathrm{PSH}(B)$ with $0\leq w\leq 1$. Since $|u_ j|\leq \varepsilon _ j|\log (|\zeta |^2/R^2)|+\eta _ j$, the weighted Chern–Levine–Nirenberg estimate applied to $\log (|\zeta |^2/R^2)$, together with the range estimate (5.27.3) for the constant term, gives

where $C_ B$ depends on the fixed balls and the logarithm but not on $j$ or $w$. Restricting the integral to $B'\cap \{ |u_ j|>t\} $, dividing by $t>0$ and taking the supremum over $w$ shows that the capacity of $B'\cap \{ |u_ j|>t\} $ relative to $B$ tends to zero.

In (5.27.30), the factor $\varepsilon _ j(\mathrm{dd}^{\mathrm{c}}v_ j)^ m$ equals $(\mathrm{dd}^{\mathrm{c}}H_{\delta _ j})^ m$. For $z'\in P$ and $|t|\leq \eta _ j$, the radii satisfy

Extend $f\in C_ c(\Omega )$ by zero to $\mathbb C^ n$. Outside the projection of $\operatorname {supp}f$ to the $z'$ variables, both $f(r_ t\mathrm e^{\mathrm i\vartheta },z')$ and $f(0,z')$ vanish. On this projection, the uniform continuity of $f$ and the bound on the radii make their difference tend to zero uniformly. Since the weights $w_{\eta _ j}(t)\, \mathrm dt$ and $\mathrm d\vartheta /(2\pi )$ and the measures $(\mathrm{dd}^{\mathrm{c}}H_{\delta _ j})^ m$ all have total mass one, it follows that

The second integral tends to $f(0)$, which proves the asserted vague limit. $\square$

Proof. Write $S=\mathrm{dd}^{\mathrm{c}}v$, $q=\varepsilon \log (|\zeta |^2/R^2)$ and $T_ A=v-A-q$. The rank-one calculation in the proof of lemma 5.27.29 gives

off $D$; since $h_ A$ is smooth and $D$ has measure zero, this formula determines the current. Fix a smooth compactly supported test form. On the projection of its support to $U$, the function $v$ and its derivatives are bounded. Hence the support of $\chi _\eta '(T_ A)$ lies in $\{ |\zeta |\leq s_ A\} $, where $s_ A=C\exp (-A/(2\varepsilon ))$ and $C$ depends only on this compact set and the fixed data. The coefficients involving $v$ and $\chi _\eta $ are bounded at the fixed parameters, so the first summand has pairing $O(s_ A^2)$. In the second summand, the term with two derivatives of $v$ also has pairing $O(s_ A^2)$, and the two cross terms have pairing $O(s_ A)$: the derivative of $q$ in the $\zeta $ direction has size $O(|\zeta |^{-1})$, and its area integral over the disk $\{ |\zeta |\leq s_ A\} $ is $O(s_ A)$.

It remains to integrate the term containing $\mathrm dq\wedge \mathrm d^{\mathrm c}q$. The logarithmic change of the radial variable used for (5.27.30) leaves the factor $\varepsilon $ and the probability weight

Only the component of the test form in the $z'$ directions enters this term, and on the shrinking circles its coefficients tend uniformly to their values at $\zeta =0$. Integration in the angle therefore gives, in the limit, exactly the pairing with $\varepsilon [D]\wedge S^{r-1}$. This proves the convergence in every degree. $\square$

Proof. Put $M=1+\| b\| _{L^\infty (L)}$. Cover $K$ by finitely many balls whose larger concentric balls have closures in $L^\circ $. Replacing $b$ by $b+M$, which has the same Hessian and takes values in $[0,2M]$ on $L$, we may assume $0\leq b\leq 2M$ there. For nested cutoff functions $\psi $, integration by parts against closed forms gives

where $W$ is $T$ multiplied by the remaining positive closed factors. Apply this successively to the $n-r$ Hessians of $b$. For bounded $b$, the integrations by parts are justified by local decreasing regularization and theorem 5.27.1, with one common bound on the range. Summing over the finite cover gives the asserted constant. $\square$

Proof. We take for $h$ a function from lemma 5.27.29 whose top-degree measure approximates $c\delta _ a$ and whose lower-degree currents are small; lemma 5.27.34 then controls the mixed terms.

Write $a=(a_1,a')$ and $m=n-1$, and fix $R=2\sup _\Omega |z_1|+2$. For $\delta >0$, put $H_\delta (z')=\log (|z'-a'|^2+\delta ^2)$. By the radial calculation in the proof of proposition 5.27.31, $(\mathrm{dd}^{\mathrm{c}}H_\delta )^ m$ is a probability measure tending to $\delta _{a'}$ as $\delta \downarrow 0$. Extend every $f_ i$ by zero outside $\Omega $, and choose $\delta $ such that

Choose compact sets $K,L\subset \Omega $ with $K\subset L^\circ $ and $K$ containing the supports of all $f_ i$, and a nonnegative smooth cutoff function $\psi $ with compact support in $\Omega $, equal to one near $L$. Put $\beta =\mathrm{dd}^{\mathrm{c}}|z|^2$. Expanding the top power of $\mathrm{dd}^{\mathrm{c}}(b+h)$, summing the bounds of lemma 5.27.34 with their binomial coefficients and the norms $\| f_ i\| _\infty $, we find $\gamma >0$ such that the mixed terms contribute less than $\tau /3$ to every test integral whenever $h$ is smooth and plurisubharmonic and

Fix $0<\eta \leq t/2$. Apply lemma 5.27.29 on $\mathbb C\times \mathbb C^ m$ with transverse variable $\zeta =z_1-a_1$ and $v=c^{1/m}\varepsilon ^{-1/m}H_\delta $. For $r<n$, lemma 5.27.33 gives

as $A\to \infty $. For the fixed $\delta $, the integrals of these limits against $\psi \beta ^{n-r}$ are finite, and their exponents of $\varepsilon $ are positive. Choose $0<\varepsilon \leq t/2$ so that each of them is less than $\gamma /2$. With these parameters fixed, choose $A$ so large that the integrals $\int _ L(\mathrm{dd}^{\mathrm{c}}h_ A)^ r\wedge \beta ^{n-r}$ are less than $\gamma $, that $v-A\leq 0$ on the projection of $\Omega $ to the $z'$ variables, and that

The last choice follows directly from (5.27.30), because the circles there shrink uniformly over the projected supports of the tests and $\varepsilon (\mathrm{dd}^{\mathrm{c}}v)^ m=c(\mathrm{dd}^{\mathrm{c}}H_\delta )^ m$.

Set $h=h_ A-\eta $. As in the proof of proposition 5.27.31, $h$ is smooth, bounded and nonpositive on $\Omega $, with

The logarithm has $L^ p(\Omega )$ norm bounded uniformly for $a_1$ in the projection of $\Omega $ to the first coordinate: after translation, its one-variable integral is bounded by polar integration on one fixed disk. This uniform bound, the weighted Chern–Levine–Nirenberg estimate and the range estimate (5.27.3) give the asserted bound for the integrals over $B'$. Since $\varepsilon +\eta \leq t$, both estimates hold with constants depending only on $\Omega $, $p$, $B'$ and $B$.

Finally, multilinearity of bounded products gives

The choice of $\delta $, the choice of $A$ for the top-degree term and the estimate of the mixed terms each contribute less than $\tau /3$ to every test integral. This proves the required inequalities, with one choice of $h$ for all finite $p$. $\square$

Proof. Choose relatively compact open sets exhausting $\Omega $, each with closure contained in the next one. On each of these sets, take a countable family of smooth compactly supported functions that is uniformly dense in the space of continuous functions vanishing at its boundary, and add a nonnegative smooth compactly supported cutoff function equal to one on its closure. Enumerate the union of these families as $(f_ i)$. If positive measures converge against every $f_ i$ to a fixed positive Radon measure, they converge vaguely: the cutoff functions give common local mass bounds, and uniform approximation then handles every continuous compactly supported test function.

For each $j$, approximate $\nu $ against $f_1,\ldots ,f_ j$ by a finite positive atomic measure

Such an approximation uses only the finite mass of $\nu $ on a compact neighborhood of the supports of these tests: a finite partition on which all the tests have small oscillation provides the atoms and their positive weights. Atoms of weight zero are omitted. If $N_ j=0$, set $u_ j=b$. Otherwise start with $b_{j,0}=b$ and apply proposition 5.27.35 successively to $b_{j,s-1}$, $a_{j,s}$ and $c_{j,s}$, with the tests $f_1,\ldots ,f_ j$ and $\tau =t=(jN_ j)^{-1}$. This gives perturbations $h_{j,s}$, and we define

Every intermediate potential is bounded and plurisubharmonic, so the proposition applies at each step; the choice at the next step may depend on the bound of the current potential, and each perturbation serves all finite $p$ at once. Summing the norm estimates yields $\| u_ j-b\| _{L^ p(\Omega )}\leq C_{\Omega ,p}/j$, and since every $h_{j,s}\leq 0$, we have $u_ j\leq b$. Summing the measure errors gives

By the choice of the family $(f_ i)$, this proves the asserted vague limit, also when $\nu (\Omega )=\infty $.

For fixed concentric balls $B'\Subset B\Subset \Omega $, the perturbations are nonpositive, so $|u_ j-b|=\sum _ s|h_{j,s}|$. The last estimate of proposition 5.27.35 bounds the integral of this function against $(\mathrm{dd}^{\mathrm{c}}w)^ n$ over $B'$ by $C_{B',B,\Omega }/j$, for every $w\in \mathrm{PSH}(B)$ with $0\leq w\leq 1$. Restricting to the set where $|u_ j-b|$ exceeds a fixed threshold, dividing by the threshold and taking the supremum over $w$ proves convergence in local capacity. If $b$ is smooth, the finite sums defining $u_ j$ are smooth as well. $\square$

Proof. Choose relatively compact open sets $U_\ell $ exhausting $\Omega $, with $\overline U_\ell \subset U_{\ell +1}$. For each $\ell $, choose a countable uniformly dense subset of $C_0(U_{\ell +1};\mathbb R)$ consisting of smooth compactly supported functions, extended by zero to $\Omega $; such families exist because $C_0(U_{\ell +1};\mathbb R)$ is separable and its elements can be approximated uniformly by smooth functions. Add a nonnegative smooth compactly supported cutoff function equal to one near $\overline U_{\ell +1}$, and enumerate the union of all these functions as $(f_ i)$.

Call the maps $\alpha \mapsto \int _\Omega f_ i\, \mathrm d\alpha $ the coordinates. If positive measures converge to a fixed measure in every coordinate, the cutoff coordinates give common mass bounds on each $\overline U_{\ell +1}$, and uniform approximation then gives convergence against every $g\in C_ c(\Omega )$. The same argument, applied near a fixed measure, shows that $\alpha \mapsto \int _\Omega g\, \mathrm d\alpha $ is continuous for the topology generated by the coordinates: bound the relevant cutoff coordinate on a neighborhood, choose one smooth test that is uniformly close to $g$, and shrink the neighborhood using the coordinate of that test. Thus the coordinates generate exactly the vague topology, and they separate measures.

The function $s\mapsto s/(1+s)$ is increasing and subadditive on $[0,\infty )$. Applying it to the triangle inequalities for the coordinates gives the triangle inequality for $d_ v$, and $d_ v$ is positive on distinct measures because the coordinates separate them. Each coordinate is continuous for this metric, and the tails of the series are uniformly small, so the metric induces exactly the topology generated by the coordinates.

Finally, the finite atomic measures with nonnegative rational weights at points of a fixed countable dense subset of $\Omega $ form a countable dense family. To approximate a measure in finitely many coordinates, partition a compact neighborhood of the supports of the corresponding tests into sets on which all these tests have small oscillation, place an atom in each set of nonzero mass, and then perturb its location and weight. Only the finite mass of that compact neighborhood is used, and the remaining tail of the metric is arbitrarily small. $\square$

Proof. In a metric space, the full cluster set of a sequence is the intersection of the closures of its tails: a point of this intersection can be approximated within $1/k$ at an index larger than the preceding one, successively for $k=1,2,\ldots $. The full cluster set is therefore closed, and lemma 5.27.38 proves necessity.

For sufficiency, suppose first that $F$ is nonempty and closed. A subset of a separable metric space is separable, so we may choose a dense sequence $\mu _1,\mu _2,\ldots $ in $F$, repetitions allowed. Concatenating the finite lists $(\mu _1)$, $(\mu _1,\mu _2)$, $(\mu _1,\mu _2,\mu _3)$, and so on, we obtain a sequence $(\nu _ j)$ in which every $\mu _ k$ appears arbitrarily late. Hence every point of $F$ is the limit of a subsequence, and since $F$ is closed, no other subsequential limit occurs. If $F$ is empty, take $\nu _ j=j\delta _ a$ for a fixed $a\in \Omega $. A nonnegative compactly supported test function equal to one at $a$ has integral $j$ against $\nu _ j$, so this sequence has no vaguely convergent subsequence.

In either case, let $(f_ i)$ and $d_ v$ be as in lemma 5.27.38, and perform the finite atomic construction from the proof of theorem 5.27.36 separately for each $j$, with background zero, target $\nu _ j$ and tests $f_1,\ldots ,f_ j$. Thus we approximate $\nu _ j$ in these coordinates by a finite atomic measure with error less than $1/j$. If this measure has $N_ j>0$ atoms, apply proposition 5.27.35 successively with $t=\tau =(jN_ j)^{-1}$; if it has none, take $u_ j=0$. The resulting smooth bounded plurisubharmonic functions $u_ j\leq 0$ satisfy

On every fixed pair of concentric balls $B'\Subset B\Subset \Omega $, the bounds of that proposition for test measures $(\mathrm{dd}^{\mathrm{c}}w)^ n$, with $w\in \mathrm{PSH}(B)$ and $0\leq w\leq 1$, also sum to $C_{B',B,\Omega }/j$. Dividing by a positive threshold and taking the supremum over $w$ gives convergence in local capacity. These estimates require neither convergence of the targets $\nu _ j$ nor a common local mass bound for them.

Splitting the series defining the metric after its first $j$ terms gives

Consequently, the two sequences of measures have the same limits along corresponding subsequences, and hence the same full cluster set $F$. $\square$

Proof. First suppose that $f$ is supported in $B'$, where $B'\Subset B\Subset \Omega $ are concentric balls. Put $v_ j=\max (u_ j,b-1)$ and $E_ j=\{ u_ j\leq b-1\} $. The functions $v_ j$ lie between $b-1$ and $b$ and tend to $b$ in local capacity. Because of their common bounds, the continuity of bounded products recalled in section 5.27 gives $(\mathrm{dd}^{\mathrm{c}}v_ j)^ n\to (\mathrm{dd}^{\mathrm{c}}b)^ n$ vaguely on $B$. With $R=\sup _ B b-\inf _ B b+1$, the capacity bound for bounded potentials gives

where the limit holds because $E_ j\subset \{ |u_ j-b|>1/2\} $. On the plurifine open Borel set $\{ u_ j>b-1\} $ we have $v_ j=u_ j$, so theorem 5.27.1 shows that the two measures agree there. By positivity,

Taking the lower limit proves the assertion for such $f$. For a general $f$, use a finite nonnegative partition of unity subordinate to a cover of its support by such balls. Applying the inequality along a vaguely convergent subsequence proves the final assertion. $\square$

Proof. If the second condition holds, lemma 5.27.41 gives the measure inequality, and the full cluster set is closed in the metrizable vague topology by the argument at the beginning of the proof of theorem 5.27.39.

Conversely, assume the first condition. If $F$ is nonempty, the concatenation of a dense sequence used in the proof of theorem 5.27.39 gives measures $\nu _ j\in F$ whose full vague cluster set is $F$. If $F$ is empty, take $\nu _ j=(\mathrm{dd}^{\mathrm{c}}b)^ n+j\delta _ a$ for a fixed $a\in \Omega $. A nonnegative compactly supported test function equal to one at $a$ has integral at least $j$ against $\nu _ j$, so this sequence has no vaguely convergent subsequence. In either case, $\nu _ j-(\mathrm{dd}^{\mathrm{c}}b)^ n$ is a positive Radon measure.

Let $(f_ i)$ and $d_ v$ be as in lemma 5.27.38. For each $j$, approximate the excess $\nu _ j-(\mathrm{dd}^{\mathrm{c}}b)^ n$ against $f_1,\ldots ,f_ j$ by a positive finite atomic measure, with each error less than $1/j$. If this measure has $N_ j>0$ atoms, perform the construction from the proof of theorem 5.27.36, starting at $b$ and using $\tau =t=(jN_ j)^{-1}$ for each atom; if it has none, take $u_ j=b$. Summing the estimates of proposition 5.27.35 gives

For every fixed pair of concentric balls $B'\Subset B\Subset \Omega $, the same construction bounds the integral of $|u_ j-b|$ over $B'$ against $(\mathrm{dd}^{\mathrm{c}}w)^ n$ by $C_{B',B,\Omega }/j$, for every $w\in \mathrm{PSH}(B)$ with $0\leq w\leq 1$. Dividing by a fixed positive threshold and taking the supremum over $w$ gives convergence in local capacity. Each $u_ j$ is bounded and lies below $b$, and it is smooth when $b$ is smooth. One sequence serves all finite $p$, and no common local mass bound on the $\nu _ j$ is needed.

The error in the first $j$ coordinates and the tail of the series yield

By the remark following lemma 5.27.38, the two sequences of measures have the same limits along corresponding subsequences. Hence their full vague cluster sets coincide. $\square$