2.2 Convexity [008C]

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2.2 Convexity

Consider $u\in PSH(X_{t},\omega_{t})$ normalised to $\sup_{X_{t}}u=0$ . Equivalently, we can cover $X_{t}$ by a bounded number of charts as before, and use the local potentials of $\omega_{\mathcal{X}}$ to represent $u$ as a collection of local plurisubharmonic (psh) functions $\{u_{\beta}\}$ with $|u_{\beta}-u|\leq C$ .

Lemma 2.2.

(Convexity) Let $\phi$ be any psh function on the open subset of $\{1<|z_{i}|<\Lambda,i=1,\ldots p,|z_{k}|<1,k=p+1,\ldots n\}\subset(\mathbb{C}^{*})^{p}\times\mathbb{C}^{n-p}$ . Then the function

\bar{\phi}(x_{1},\ldots x_{n})=\frac{1}{(2\pi)^{n}}\int_{D(1)^{n-p}}\prod_{p+1}^{n}\sqrt{-1}dz_{k}\wedge d\bar{z}_{k}\int_{T^{p}}\phi(e^{x_{1}+i\theta_{1}},\ldots e^{x_{p}+i\theta_{p}})d\theta_{1}\ldots d\theta_{p}

is convex.

Proof.

For any choice of $\theta_{i}$ the function $\phi(z_{1}e^{i\theta_{1}},\ldots z_{p}e^{i\theta_{n}},z_{p+1},\ldots,z_{n})$ is psh, since the $T^{p}$ -action on $(\mathbb{C}^{*})^{p}$ is holomorphic. Thus the average function $\bar{\phi}$ is also psh as a function of $z_{1},\ldots z_{p}$ . Any $T^{p}$ -invariant psh function must be convex in the log coordinates, because for $x_{i}=\log|z_{i}|$ ,

\sqrt{-1}\partial\bar{\partial}\bar{\phi}=\frac{1}{4}\sum\frac{\partial^{2}\bar{\phi}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log z_{i}\wedge d\overline{\log z_{j}}\geq 0.

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