2.1 Skoda inequality [00T8]

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2.1 Skoda inequality

An upper semicontinuous $L^{1}$ -function $\phi$ on a coordinate ball is called plurisubharmonic (psh) if $\sqrt{-1}\partial\bar{\partial}\phi\geq 0$ . The basic intuition is that regularity properties for psh functions in general dimensions are analogous to subharmonic functions on Riemann surfaces. This is captured by the basic version of the Skoda inequality:

Theorem 2.1.

(cf. [40, Thm 3.1]) If $\phi$ is psh on $B_{2}\subset\mathbb{C}^{n}$ , with $\int_{B_{2}}|\phi|\omega_{E}^{n}\leq 1$ with respect to the standard Euclidean metric $\omega_{E}$ , then there are dimensional constants $\alpha$ , $C$ , such that

\log\int_{B_{1}}e^{-\alpha\phi}\omega_{E}^{n}\leq C.

Remark 2.2.

If instead $\int_{B_{2}}|\phi|\omega_{E}^{n}\leq C^{\prime}$ for some constant $C^{\prime}$ , then we can apply Thm 2.1 to a scaling of $\phi$ , to get a Skoda inequality with modified $\alpha,C$ .

Remark 2.3.

Assuming an $L^{1}$ -bound on $\phi$ , then we can take a suitable cutoff function $\chi$ , and via integration by parts,

\int_{B_{1}}\sqrt{-1}\partial\bar{\partial}\phi\wedge\omega_{E}^{n-1}\leq\int_{B_{2}}\chi\sqrt{-1}\partial\bar{\partial}\phi\wedge\omega_{E}^{n-1}=\int_{B_{2}}\phi\sqrt{-1}\partial\bar{\partial}\chi\wedge\omega_{E}^{n-1}\leq\left\lVert\chi\right\rVert_{C^{2}}\left\lVert\phi\right\rVert_{L^{1}}\leq C.

This simple idea is a basic version of the Chern-Levine inequality, which is another fundamental reason why psh functions are much more regular than the subharmonic functions in general dimensions.

The basic Skoda inequality immediately implies a global version. On a compact Kähler manifold $(X,\omega)$ , we say an upper semicontinuous $L^{1}$ -function $\phi\in PSH(X,\omega)$ if $\omega_{\phi}=\omega+\sqrt{-1}\partial\bar{\partial}\phi\geq 0$ . This is the generalised notion of Kähler potentials.

Theorem 2.4.

On a fixed $(X,\omega)$ , there are positive constants $\alpha$ , $C$ depending only on $X,\omega$ , such that

\int_{X}e^{-\alpha\phi}\omega_{X}^{n}\leq C,\quad\forall\phi\in PSH(X,\omega)\text{ with }\sup\phi=0.

Remark 2.5.

Here $\int_{X}|\phi|\omega_{X}^{n}$ is automatically bounded using the Harnak inequality, because $\Delta\phi\geq-n$ for $\phi\in PSH(X,\omega)$ .

Remark 2.6.

The supremum of all such $\alpha$ is known as Tian’s alpha invariant.