5.7 Plurisubharmonic functions7B39B08A9568

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Plurisubharmonic functions are the basic objects of pluripotential theory. On a domain $\Omega \subset \mathbb {C}^ n$, a plurisubharmonic function is an upper semicontinuous function whose restriction to every complex line is subharmonic. By (5.2.9), the Laplacian of a function on $\Omega $, viewed as an open subset of $\mathbb R^{2n}$, is four times the trace of its complex Hessian, so subharmonicity controls only this trace. Subharmonicity on all complex lines is the pointwise counterpart of the positivity of the whole complex Hessian, $\mathrm{dd}^{\mathrm{c}}u\geq 0$.

Since $\mathrm{dd}^{\mathrm{c}}u\geq 0$ determines $u$ only almost everywhere, the pointwise definition and the distributional condition must be kept apart. We first show that the upper semicontinuity required in the definition follows from the condition on complex lines (theorem 5.8.4); the proof combines the incidence estimate of section 2.9 with the distributional theory of section 3.4. We then identify the plurisubharmonic functions with the canonical representatives, in the sense of theorem 3.4.12, of the distributions with positive complex Hessian (theorem 5.9.1). This makes the potential theory of Part I available. A smooth function is plurisubharmonic exactly when its complex Hessian is positive semidefinite, and mollification approximates every plurisubharmonic function from above by smooth ones. Plurisubharmonicity is invariant under holomorphic maps, hence makes sense on complex manifolds, and it is preserved by decreasing limits, regularized upper envelopes and $L^1_{\mathrm{loc}}$ limits. Moreover, plurisubharmonic functions belong to $L^ p_{\mathrm{loc}}$ for every finite $p$, and near each point $\mathrm e^{-\alpha u}$ is integrable for small $\alpha >0$.

In the last two sections we extend plurisubharmonic functions across closed sets that are locally contained in zero sets of holomorphic functions or are small in the sense of Hausdorff measure, and we prove Kiselman's minimum principle: although an infimum of plurisubharmonic functions is in general not plurisubharmonic, a partial infimum over real variables is plurisubharmonic or identically $-\infty $ under convexity assumptions, provided that the function does not depend on the corresponding imaginary parts.

Historical remarks and references for this chapter are collected in subsection 25.1.8.

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Plurisubharmonic functions are studied through their restrictions to complex lines, their positive complex Hessians, and their local integrability. The conventions and normalizations below also apply to quasi-plurisubharmonic functions and singular Hermitian metrics.

The chapter begins with plurisubharmonic functions on domains in $\mathbb C^ n$ and on complex manifolds. We also include the $0$-dimensional case, which is occasionally convenient in inductive arguments. The basic permanence properties of plurisubharmonic functions are recalled: stability under upper envelopes and decreasing limits, Choquet's lemma, local integrability, gluing, extension across analytic subsets, and Kiselman's principle.

The plurifine topology makes local plurisubharmonic functions continuous. Its neighborhood bases and local connectedness are used in arguments involving contact sets and non-pluripolar products.

The next part of the chapter fixes our conventions for Lelong numbers and multiplier ideal sheaves. These conventions differ by harmless but important factors from some common references; the conversions are explained in 5.16.3, 5.16.12. We also recall Siu's semicontinuity theorem, the strong openness theorem of Guan–Zhou, the Ohsawa–Takegoshi extension theorem, and the valuative characterization of multiplier ideals.

The remaining sections develop the global language used throughout these constructions. We introduce $\theta $-plurisubharmonic functions, quasi-plurisubharmonic functions, the preorder $\preceq $ of singularity types, and compactness properties in $L^1$. We then recall analytic singularities, log resolutions and quasi-equisingular approximation. Finally, we discuss closed positive $(1,1)$-currents, pseudo-effective and big classes, Kähler currents, Siu decomposition, non-Kähler and non-nef loci, modified nef classes, and singular Hermitian metrics on line bundles.

Nets and subnets use the convention in 2.7 Nets and compactness. In particular, a subnet of a monotone net is monotone.