4.2 Complex pluripotential theory [004C]

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4.2 Complex pluripotential theory

Complex pluripotential theory concerns the study of weak notions of Kähler potentials, and their wider implications on algebraic geometry and PDEs. They can be viewed as the generalization of potential theory on Riemann surfaces to several complex variables. Some common themes include:

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The weak compactness theory for the space of potentials. This is suited for variational methods in Kähler geometry.
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The relations between potential theory and algebraic geometry, via Hodge theory, $\bar{\partial}$ -operator, intersection theory, etc. This provides transcendental methods to birational geometry.
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The weak formulation of the complex Monge-Ampère equation, and a priori potential estimates under weak assumptions on the volume density, or in the presence of singularities for the ambient complex variety. This is useful for studying singularity formation of canonical metrics.

Unlike Yau’s proof of the Calabi conjecture, whose techniques are typical of elliptic PDEs, complex pluripotential theory is more akin to complex analysis, and frequently provides stronger results. To build up the intuition, we will first review the standard versions in the literature, before stating our uniform versions, which are foundational to our approach to the SYZ conjecture.

4.2.1 Skoda inequality

An upper semicontinuous $L^{1}$ -function $\phi$ on a coordinate ball in $\mathbb{C}^{n}$ is called plurisubharmonic (psh) if it satisfies the sub mean value inequality when restricted to complex lines; this implies $dd^{c}\phi\geq 0$ . 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 local version of the Skoda inequality:

Theorem 4.2.

(cf. [79, 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 5.

The Skoda inequality might be contrasted with subharmonic functions on the unit ball in $\mathbb{R}^{2n}$ for $n>1$ , for which exponential integrability is far too much to expect.

On a compact Kähler manifold $(Y,\omega)$ , we say an upper semicontinuous $L^{1}$ -function $\phi\in PSH(Y,\omega)$ if its sum with the local potential of $\omega$ is psh, so that $\omega_{\phi}=\omega+dd^{c}\phi\geq 0$ . This is the generalised notion of Kähler potentials. The standard global analogue of the Skoda inequality is:

Theorem 4.3.

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

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

In our applications, we need to work with a polarized algebraic degeneration of Calabi-Yau manifolds $\pi:X\to S\setminus\{0\}$ near the large complex structure limit, as in section 3. Let $\omega_{FS}$ be a fixed Fubini-Study metric on $(X,c_{1}(L))$ induced by a projective embedding via the sections of a high power of $L$ , and use $\omega_{FS,t}=\frac{1}{|\log|t||}\omega_{FS}|_{X_{t}}$ to define a family of background metrics on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(L)$ . The normalization factor $\frac{1}{|\log|t||}$ is to ensure two different choices of Fubini-Study reference metrics would differ by a potential with $C^{0}$ norm of order $O(1)$ independent of small $t$ . Recall $d\mu_{t}$ is the normalized Calabi-Yau measure.

We adapted the Skoda inequality to a uniform version [51]:

Theorem 4.4.

(Uniform Skoda estimate) There are uniform positive constants $\alpha,A$ independent of $t$ for $0<|t|\ll 1$ , such that for the normalised Calabi-Yau measures $d\mu_{t}$ ,

\int_{X_{t}}e^{-\alpha u}d\mu_{t}\leq A,\quad\forall u\in PSH(X_{t},\omega_{FS,t})\text{ with }\sup_{X_{t}}u=0.

The proof involves covering $X_{t}$ by plenty of small regions which look like standard balls in $\mathbb{C}^{n}$ . An elementary but somewhat tricky construction of test function allows one to estimate $L^{1}$ norms of the local potentials. One then applies the local version of Skoda inequality to each small region, and sum over all regions.