5.46 Polar curvatureAFA07BFD0E7B

Proof. Fix a smooth function $0\leq \psi \leq 1$ with compact support in $U$, and a relatively compact open neighborhood $V$ of its support. Mollify $w$ inside $V$ with a standard mollifier $\rho _\varepsilon $, writing $w_\varepsilon =w*\rho _\varepsilon $ and $\nu _\varepsilon =(\Delta w)*\rho _\varepsilon $. For small $\varepsilon $, integration by parts and Young's inequality give

Absorbing the middle term on the right, we obtain a bound for $\int \psi ^2|\nabla w_\varepsilon |^2$ that is independent of $\varepsilon $. Taking $\psi =1$ on a prescribed smaller compact set, weak compactness in $W^{1,2}$ and convergence of the mollifications in $L^2$ identify the weak limit with $w$. This proves the assertion. $\square$

Proof. Adding a smooth quadratic function and subtracting a constant, we obtain a negative plurisubharmonic function $h$ on $U$ with the same pole set as $v$. Put $h_ M=\max (h,-M)$ for $M\geq 1$, and choose $\psi \in C^\infty _ c(O)$ with $0\leq \psi \leq 1$ and $\psi =1$ near $K$. On a fixed compact neighborhood of $\operatorname {supp}\psi $, the positive measures $\Delta h_ M$ have uniformly bounded mass: choose a nonnegative smooth function $\chi $ with compact support in $U$, equal to one on this neighborhood, and use

Apply the integration-by-parts estimate of the preceding proof to mollifications of $h_ M$. Their absolute values are at most $M$, and their squares are at most $M$ times their absolute values. The uniform bounds for the mass of the Laplacians and for the local $L^1$ norms therefore give, after passing to the weak limit in $W^{1,2}$,

Here $C$ depends on the fixed cutoffs and the local $L^1$ norm of $h$, but not on $M$. The second inequality also uses $h_ M^2\leq M|h|$ in the terms containing $\psi $ and $\nabla \psi $.

The Sobolev function $-\psi h_ M/M$ takes values between zero and one. It equals one on a neighborhood of $K$, since $h$ is upper semicontinuous and therefore $\{ h<-M\} $ is an open neighborhood of $K$. Extend it by zero and mollify with a sufficiently small radius: the support stays in $O$, the function remains equal to one on a smaller neighborhood of $K$, and its Sobolev norm does not increase. Letting $M\to \infty $ gives the required smooth functions. $\square$

Proof. The two local inequalities show that the pole sets agree. Off $S$ put $w=u-v$, and give $w$ arbitrary bounded values on $S$. Since $S$ has Lebesgue measure zero, this defines the locally bounded distribution $u-v$. The coefficient measures of $\mathrm{dd}^{\mathrm{c}}w$ are locally finite, since after a smooth correction the complex Hessians of $u$ and $v$ are positive currents of order zero. Taking the Euclidean trace shows that $\Delta w$ is a signed Radon measure, so lemma 5.46.1 gives $w\in W^{1,2}_{\mathrm{loc}}$.

Consequently, the real and imaginary parts of the coefficient measures of $\mathrm{dd}^{\mathrm{c}}w$ are continuous linear functionals on $W^{1,2}_0$ of a smaller ball, since a second distributional derivative of $w$ is a first derivative of its $L^2$ gradient. Let $\nu $ be any one of these real signed measures, with Jordan decomposition $\nu =\nu ^+-\nu ^-$. Choose a compact set $K$ contained in the intersection of $S$ with a positive set of a Hahn decomposition, so that $\nu ^-(K)=0$. For $\epsilon >0$, take an open neighborhood $O$ of $K$ in the ball with $\nu ^-(O)<\epsilon $. The test functions $\varphi _ j$ of lemma 5.46.2, applied to $u$, satisfy

By continuity on the Sobolev space, the first term tends to zero. Letting $\epsilon \downarrow 0$ and using inner regularity gives $\nu ^+(S)=0$ locally, and the same argument applied to $-\nu $ gives $\nu ^-(S)=0$. Thus all coefficient measures of $\mathrm{dd}^{\mathrm{c}}w$ vanish on $S$, which proves the first identity. Since smooth forms also give $S$ zero mass, the second identity follows. $\square$

Proof. On a small coordinate ball, take plurisubharmonic local potentials $a,b$ with $T=\mathrm{dd}^{\mathrm{c}}a$ and $S=\mathrm{dd}^{\mathrm{c}}b$, by Theorem 5.4.3. Put $h=a-b\in L^1_{\mathrm{loc}}$. The trace identity gives $\mathrm{dd}^{\mathrm{c}}h\wedge \eta ^{n-1}=0$ in the sense of distributions. Dividing by the smooth positive volume form $\eta ^ n$ gives a scalar second-order operator with smooth coefficients whose principal matrix is a positive multiple of the inverse of the Hermitian matrix of $\eta $; it is uniformly elliptic on every smaller ball. Local elliptic regularity therefore makes $h$ smooth. Thus $T-S=\mathrm{dd}^{\mathrm{c}}h$ is smooth and carried by a Lebesgue null set, so its coefficient measures vanish. This proves equality locally, and hence globally. $\square$

Proof. By proposition 5.43.18, $u\circ \pi \in \mathrm{PSH}(Y,\pi ^*\theta )$. Change of variables on the locus where $\pi $ is an isomorphism, whose complement has Lebesgue measure zero in both manifolds, gives $\pi _*(u\circ \pi )=u$ as distributions and $\pi _*\pi ^*\theta =\theta $ as currents. Since pushforward by a proper map commutes with $\mathrm{dd}^{\mathrm{c}}$, we get $\pi _*T_ Y=T$.

For any Borel set $B\subset X$, the coefficient measures of $\pi _*T_ Y$ are pushforwards of the coefficient measures of $T_ Y$, multiplied by the smooth coefficients that arise from pulling back a test form. Restriction therefore gives $\pi _*(\mathbf1_{\pi ^{-1}(B)}T_ Y)=\mathbf1_ B\pi _*T_ Y$. Applying this to the Borel set $S$ proves (5.46.6); closedness of the restricted currents is not used. $\square$

Proof. Locally, adding a smooth quadratic function to $u$ and subtracting a constant gives a negative plurisubharmonic function $a$ with pole set $S=\{ u=-\infty \} $. Put $\Theta =\mathrm{dd}^{\mathrm{c}}a$. Since smooth forms give $S$ zero mass, the polar current equals $\mathbf1_ S\Theta $ and is positive. In dimension one it is closed for degree reasons.

Now suppose that $n\geq 2$. Choose a smooth convex nondecreasing function $\chi $ on $\mathbb R$ with $\chi (t)=0$ for $t\leq 1/2$ and $\chi (1)=1$, and a smooth function $0\leq \vartheta \leq 1$ that vanishes near zero and equals one near one. Set

These functions are bounded and plurisubharmonic, with $0\leq w_ k\leq 1$. They vanish on $S$ and tend to one at every point outside $S$. Hence $\vartheta (w_ k)\Theta $ converges locally, coefficientwise in total variation, to $\mathbf1_{X\setminus S}\Theta $. Moreover, $\vartheta '(w_ k)$ vanishes on $S$ and tends to zero everywhere.

Fix a constant strongly positive $(n-2,n-2)$-form $\beta $ and put $\Xi =\Theta \wedge \beta $. The bounded Bedford–Taylor chain rule gives

Indeed, approximate $w_ k$ on a smaller ball by a decreasing sequence of smooth plurisubharmonic functions. Dominated convergence gives convergence of the corresponding $\vartheta $-weighted currents, while the convergence of bounded Bedford–Taylor energies and the Cauchy–Schwarz inequality give convergence of the differentiated expressions; this proves the chain rule for $w_ k$. For a smooth compactly supported test one-form $\alpha $, choose a nonnegative smooth cutoff $\tau $ equal to one near its support. Since $w_ k\geq 0$ and $\Xi $ is closed and positive, bounded integration by parts gives

Here $C$ depends only on the local mass of $\Xi $ and the fixed cutoff. The Cauchy–Schwarz inequality for a positive current now yields

where $\kappa $ is the Euclidean form; the limit follows by dominated convergence with respect to the fixed locally finite trace measure. Hence $\mathrm d(\vartheta (w_ k)\Xi )\to 0$, and $\mathbf1_{X\setminus S}\Theta \wedge \beta $ is closed. Since constant strongly positive forms span all constant forms of bidegree $(n-2,n-2)$, the current $\mathbf1_{X\setminus S}\Theta $ is closed. Subtracting it from $\Theta $ shows that $\mathbf1_ S\Theta $ is closed. The local currents agree on overlaps and give the asserted global current. $\square$