Plurisubharmonic potentials

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

On an open subset of $\mathbf C^n$, a real function $\varphi$ determines the two-form $dd^c\varphi$. When $\varphi$ is plurisubharmonic, the form $dd^c(|z|^2+\varphi)$ is symplectic, and the balls of the next lesson are found in such forms. This lesson proves the facts about these potentials that the course uses: their behaviour under holomorphic maps and on complex lines (Lemmas 3.1 to 3.3), the potentials $|f|^2$ and $\log|f|^2$ of holomorphic maps (Lemma 3.5), a smooth maximum (Lemma 3.7), and radial potentials (Lemma 3.9). The conventions follow [OpenAI-SB, Section 2]. Forms, pullbacks and exterior derivatives are those of Forms, flows and Moser's method.

1. The operator $d^c$

Identify $\mathbf C^n$ with $\mathbf R^{2n}$ by writing $z=u+ix$ with $u,x\in\mathbf R^n$. Multiplication by $i$ acts on real tangent vectors by $\partial_{u_j}\mapsto\partial_{x_j}$ and $\partial_{x_j}\mapsto-\partial_{u_j}$. A map $G$ from an open subset of $\mathbf C^n$ to $\mathbf C^{n'}$ is holomorphic if it is smooth and its derivative is complex-linear at every point, $DG(iX)=i\,DG(X)$. For $n=n'=1$ these are the holomorphic functions of one variable, which are infinitely differentiable by Theorem 3.3.3 of Lebl's guide in the core course Complex Analysis; compositions of holomorphic maps are holomorphic.

For a smooth real function $\varphi$, define the one-form $d^c\varphi$ and the two-form $dd^c\varphi$ by $\begin{aligned} d^c\varphi(X)&=-\tfrac14\,d\varphi(iX),\\ dd^c\varphi&=d(d^c\varphi). \end{aligned}$ In coordinates, $d^c\varphi=\tfrac14\sum_j\bigl(\partial_{u_j}\varphi\,dx_j-\partial_{x_j}\varphi\,du_j\bigr).$ For $\varphi=|z|^2$ this gives the standard primitive and form $\begin{aligned} \lambda_0&=d^c|z|^2=\tfrac12\sum_j(u_j\,dx_j-x_j\,du_j),\\ \omega_0&=dd^c|z|^2=\sum_jdu_j\wedge dx_j. \end{aligned}$ With $(q,p)=(u,x)$ this is the standard symplectic form of the first lesson, and $B^{2n}(c)=\{z:\pi|z|^2<c\}$. Every $dd^c\varphi$ is closed (Exercise 2.1), and $d^c(\psi\circ\varphi)=\psi'(\varphi)\,d^c\varphi$ for a smooth function $\psi$ of one real variable, by the chain rule.

Lemma 3.1. Let $G$ be holomorphic and $\varphi$ smooth on the target. Then $G^*(d^c\varphi)=d^c(\varphi\circ G)$ and $G^*(dd^c\varphi)=dd^c(\varphi\circ G)$.

Proof. For a tangent vector $X$, $(G^*d^c\varphi)(X)=d^c\varphi(DG(X))$, which is $-\frac14d\varphi(i\,DG(X))$, and $i\,DG(X)=DG(iX)$, so this equals $-\frac14d(\varphi\circ G)(iX)$. The second identity follows by Lemma 2.1(b). $\square$

Lemma 3.2. In one variable $\zeta=s+it$, $dd^ch=\frac14\Delta h\,ds\wedge dt$ for every smooth real $h$. If $\phi$ is holomorphic, then $\Delta|\phi|^2=4|\phi'|^2$ and $dd^c|\phi|^2=|\phi'|^2\,ds\wedge dt$.

Proof. $d^ch=\frac14(h_s\,dt-h_t\,ds)$, and $d$ of this is $\frac14(h_{ss}+h_{tt})\,ds\wedge dt$. For $\phi=a+ib$, the functions $a,b$ are harmonic and satisfy $|\nabla a|^2=|\nabla b|^2=|\phi'|^2$ by the Cauchy–Riemann equations; so $\Delta(a^2+b^2)=2|\nabla a|^2+2|\nabla b|^2=4|\phi'|^2$. $\square$

Lemma 3.3 (complex lines). For a smooth real function $\varphi$ near $z\in\mathbf C^n$ and $X\in\mathbf C^n$, $dd^c\varphi(z)(X,iX)=\tfrac14\,\Delta_\zeta\,\varphi(z+\zeta X)\big|_{\zeta=0},$ the Laplacian taken in the complex variable $\zeta$.

Proof. Apply Lemma 3.1 to the holomorphic map $G(\zeta)=z+\zeta X$ and Lemma 3.2 to $\varphi\circ G$. At $\zeta=0$ the vectors $\partial_s$, $\partial_t$ are mapped to $X$ and $iX$. $\square$

2. Plurisubharmonic functions

A smooth real function $\varphi$ on an open set of $\mathbf C^n$ is plurisubharmonic if $dd^c\varphi(X,iX)\ge0$ for all points and all tangent vectors $X$, that is, if its restrictions to complex lines are subharmonic in the sense $\Delta_\zeta\varphi(z+\zeta X)\ge0$.

Lemma 3.4. If $\varphi$ is plurisubharmonic, then $\omega=dd^c(|z|^2+\varphi)$ is a symplectic form with $\omega(X,iX)\ge|X|^2$.

Proof. The form is closed. For $X=(a,b)$ in the coordinates $(u,x)$, $iX=(-b,a)$ and $\omega_0(X,iX)=|a|^2+|b|^2=|X|^2$. Hence $\omega(X,iX)\ge|X|^2$, and $\omega(X,\cdot)=0$ forces $X=0$. $\square$

For holomorphic functions of one variable we use the operators $\partial_\zeta=\frac12(\partial_s-i\partial_t)$ and $\partial_{\bar\zeta}=\frac12(\partial_s+i\partial_t)$, for which $\Delta=4\partial_\zeta\partial_{\bar\zeta}$, $\partial_{\bar\zeta}h=0$ and $\partial_\zeta h=h'$ if $h$ is holomorphic, and $\partial_{\bar\zeta}\bar h=\overline{h'}$.

Lemma 3.5. Let $f=(f_1,\dots,f_m)$ be holomorphic on an open set $\Omega\subseteq\mathbf C^n$ and $\tau=|f|^2=\sum_j|f_j|^2$. Then, with $df(X)\in\mathbf C^m$ the derivative of $f$ along $X$, $dd^c\tau(X,iX)=|df(X)|^2\ge0,$ and, where $f\ne0$, $\begin{aligned} &dd^c\log\tau(X,iX)\\ &\quad=\frac{|f|^2|df(X)|^2-\bigl|\sum_j\overline{f_j}\,df_j(X)\bigr|^2}{|f|^4}\\ &\quad\ge0. \end{aligned}$ So $\tau$ is plurisubharmonic, and so is $\log\tau$ where $f$ has no zero.

Proof. Fix $z$, $X$ and put $h(\zeta)=f(z+\zeta X)$, a holomorphic map of one variable with $h'(0)=df(X)$. By Lemma 3.3 we need $\frac14\Delta$ of $|h|^2$ and of $\log|h|^2$ at $0$. First $\partial_{\bar\zeta}|h|^2=\sum_jh_j\overline{h_j'}$ and $\partial_\zeta\partial_{\bar\zeta}|h|^2=|h'|^2$. Next $\begin{aligned} \partial_{\bar\zeta}\log|h|^2&=\frac{\sum_jh_j\overline{h_j'}}{|h|^2},\\ \partial_\zeta\partial_{\bar\zeta}\log|h|^2&=\frac{|h'|^2}{|h|^2}-\frac{\bigl|\sum_jh_j\overline{h_j'}\bigr|^2}{|h|^4}. \end{aligned}$ The numerator $|h|^2|h'|^2-|\sum_j\overline{h_j}h_j'|^2$ is nonnegative by the Cauchy–Schwarz inequality. $\square$

Lemma 3.6. Let $\Gamma$ be a smooth function on an open subset of $\mathbf R^N$ that is convex and nondecreasing in each argument, and let $h_1,\dots,h_N$ be plurisubharmonic with $(h_1,\dots,h_N)$ in the domain of $\Gamma$. Then $\Gamma(h_1,\dots,h_N)$ is plurisubharmonic.

Proof. On a complex line, with the subscripts of $\Gamma$ denoting partial derivatives and $\nabla$ the gradient in $(s,t)$, $\begin{aligned} \Delta\,\Gamma(h)&=\sum_j\Gamma_j\,\Delta h_j\\ &\quad+\sum_{i,j}\Gamma_{ij}\,\langle\nabla h_i,\nabla h_j\rangle. \end{aligned}$ The first sum is nonnegative because $\Gamma_j\ge0$ and $\Delta h_j\ge0$; the second is the sum over the two real directions of a quadratic form in the positive semidefinite Hessian of $\Gamma$. $\square$

Lemma 3.7 (smooth maximum). Let $\vartheta$ be a smooth even nonnegative function supported in $(-1,1)$ with $\int\vartheta=1$, and put $\begin{aligned} \rho(s)&=\int_{\mathbf R}|s-t|\,\vartheta(t)\,dt,\\ \operatorname{rmax}(h,p)&=\tfrac12\bigl(h+p+\rho(h-p)\bigr). \end{aligned}$ Then $\rho$ is smooth and convex, $|\rho'|\le1$, and $\rho(s)=|s|$ for $|s|\ge1$. Hence $\operatorname{rmax}$ is smooth, convex, nondecreasing in each argument, and equal to $\max(h,p)$ when $|h-p|\ge1$.

Proof. Splitting the integral at $t=s$ gives $\rho'(s)=2\Theta(s)-1$ with $\Theta(s)=\int_{-\infty}^s\vartheta$, a smooth function with values in $[0,1]$; so $\rho$ is smooth, $|\rho'|\le1$, and $\rho''=2\vartheta\ge0$. For $s\ge1$, $|s-t|=s-t$ on the support of $\vartheta$, and $\int t\vartheta(t)\,dt=0$ by evenness, so $\rho(s)=s$; likewise $\rho(s)=-s$ for $s\le-1$. The partial derivatives of $\operatorname{rmax}$ are $\frac12(1\pm\rho'(h-p))\ge0$, and convexity is inherited from $\rho$. $\square$

3. Two explicit potentials

Lemma 3.8. Let $b\in\mathbf R^n$, $\ell(z)=\sum_jb_jz_j$, and let $\phi$ be holomorphic on an open set $D\subseteq\mathbf C$ with coordinate $\zeta=v+it$. On $\{z:\ell(z)\in D\}$, $\begin{aligned} &dd^c|\phi\circ\ell|^2\\ &\quad=|\phi'(\ell(z))|^2\;d\langle b,u\rangle\wedge d\langle b,x\rangle. \end{aligned}$

Proof. The map $\ell$ is complex-linear, hence holomorphic, and $\ell(z)=\langle b,u\rangle+i\langle b,x\rangle$ because $b$ is real. By Lemmas 3.1 and 3.2, $dd^c|\phi\circ\ell|^2=\ell^*(|\phi'|^2\,dv\wedge dt)$, and $\ell^*dv=d\langle b,u\rangle$, $\ell^*dt=d\langle b,x\rangle$. $\square$

Lemma 3.9 (radial potentials). Let $r>0$ and let $\Psi$ be smooth on an open interval containing $[0,r^2]$, with $\Psi'>0$ and $\frac d{dt}\bigl(t\Psi'(t)\bigr)>0$ there. Put $R(z)=\sqrt{\Psi'(|z|^2)}\,z$ on $B_r=\{|z|<r\}$. Then $R^*\lambda_0=d^c\Psi(|z|^2)$ and $R^*\omega_0=dd^c\Psi(|z|^2)$, and $R$ is a diffeomorphism of $B_r$ onto the ball $B^{2n}(\pi r^2\Psi'(r^2))$.

Proof. Write $R=\varrho z$ with $\varrho=\sqrt{\Psi'(|z|^2)}$. Then $\begin{aligned} R^*\lambda_0&=\tfrac12\sum_j\bigl(\varrho u_j\,d(\varrho x_j)\\ &\qquad-\varrho x_j\,d(\varrho u_j)\bigr)=\varrho^2\lambda_0, \end{aligned}$ the terms containing $d\varrho$ cancelling, and $\varrho^2\lambda_0=\Psi'(|z|^2)\,d^c|z|^2=d^c\Psi(|z|^2)$. Lemma 2.1(b) gives $R^*\omega_0=d(R^*\lambda_0)=dd^c\Psi(|z|^2)$. The map $R$ is smooth, also at $0$, and maps the sphere $|z|=\sqrt t$ onto the sphere of squared radius $t\Psi'(t)$, which increases strictly from $0$ to $r^2\Psi'(r^2)$; so $R$ is a bijection of $B_r$ onto that ball. Its derivative at $z\ne0$ is multiplication by $\varrho$ on vectors orthogonal to $z$ and by $\frac d{d\sqrt t}\sqrt{t\Psi'(t)}>0$ in the direction of $z$, and $DR(0)=\sqrt{\Psi'(0)}\,I$; by Lemma 2.2, $R$ is a diffeomorphism. $\square$

4. Exercises

3.1 (easy). Compute $d^c\varphi$ and $dd^c\varphi$ for $\varphi=|z_1|^2$ on $\mathbf C^2$, and check Lemma 3.3 for it.

3.2 (easy). Show that $\log(|z|^2+\delta)$ is plurisubharmonic on $\mathbf C^n$ for $\delta>0$, using Lemma 3.5.

3.3 (medium). Show that $(\operatorname{Re}z_1)^2$ and $\operatorname{Re}(z_1^2)$ are plurisubharmonic on $\mathbf C^n$, and that $-(\operatorname{Re}z_1)^2$ is not.

3.4 (easy). For $\Psi(t)=t+\mu\log(t+\delta)$ with $\mu,\delta>0$, check the hypotheses of Lemma 3.9 on every interval $[0,r^2]$.

5. Solutions

3.2. $|z|^2+\delta=|f|^2$ for the holomorphic map $f=(\sqrt\delta,z_1,\dots,z_n)$, which has no zero.

3.3. For $\varphi=(\operatorname{Re}z_1)^2=u_1^2$, the restriction to a complex line is $\bigl(\operatorname{Re}(z_1+\zeta X_1)\bigr)^2$, the square of a real-linear function of $\zeta$ with gradient of length $|X_1|$; its Laplacian is $2|X_1|^2\ge0$. The restriction of $\operatorname{Re}(z_1^2)$ is the real part of a holomorphic function of $\zeta$, hence harmonic, so $dd^c\operatorname{Re}(z_1^2)=0$. The restrictions of $-u_1^2$ have Laplacian $-2|X_1|^2<0$ when $X_1\ne0$.

3.4. $\Psi'(t)=1+\mu/(t+\delta)>0$ and $\frac d{dt}(t\Psi'(t))=1+\mu\delta/(t+\delta)^2>0$ on $(-\delta,\infty)$.

References