1.1 ConventionsDE921263

Here, we adopt the following conventions:

• A complex space is always assumed to be reduced, paracompact and Hausdorff.

• A modification of a complex space $X$ is a proper bimeromorphic morphism $\pi \colon Y\rightarrow X$ that is locally obtained from a finite composition of blow-ups with smooth centers.

• A subnet of a net refers to a Willard subnet. More precisely, if $f\colon I\rightarrow X$ and $f'\colon I'\rightarrow X$ are nets, namely, maps from directed sets $I$ and $I'$ to the same set $X$, we say $f'$ is a subnet of $f$ if there is an increasing final function $h\colon I'\rightarrow I$ such that $f\circ h=f'$.

• A domain in $\mathbb {R}^ d$ or $\mathbb {C}^ n$ refers to a connected open subset.

• A complex manifold is assumed to be paracompact.

• A submanifold of a complex manifold means a closed complex submanifold.

• A neighborhood is not necessarily open.

• The set $\mathbb {N}$ of natural numbers includes $0$.

• Increasing functions and decreasing functions are not necessarily strictly monotone.

• Positive forms are not necessarily strictly positive.

• The function $\mathbb {R}\ni x \mapsto x^2$ is convex instead of concave.

• We say a positive measure $\mu $ on a measurable space $X$ is carried by a measurable subset $Y$ if $\mu (X\setminus Y)=0$.

• $\mathrm{dd}^{\mathrm{c}}$ means $(2\pi )^{-1}\mathrm{i}\partial \overline{\partial }$.

We also use the following notation throughout these constructions.

• The indicator function of a set $E$ is denoted by $\mathbf1_ E$.

• For subsets $A$ and $B$ of a topological space, $A\Subset B$ means that the closure of $A$ is a compact subset of $B$; we then say that $A$ is relatively compact in $B$.

• The closure, interior and boundary of a set $E$ in its ambient space are denoted by $\overline E$, $\operatorname{Int}E$ and $\partial E$. The support of a function, measure, distribution or current is denoted by $\operatorname{Supp}$.

• For an open set $\Omega $ in $\mathbb R^ d$ or in a manifold, $C^ k(\Omega )$ is the space of $k$ times continuously differentiable functions, where $0\leq k\leq \infty $, and $C^ k_ c(\Omega )$ is its subspace of functions with compact support. We write $L^ p_{\mathrm{loc}}(\Omega )$ for the functions whose restrictions to compact subsets of $\Omega $ are $p$-integrable.

• The imaginary unit is $\mathrm{i}$, and $\operatorname{Re}z$ and $\operatorname{Im}z$ are the real and imaginary parts of a complex number $z$.

Here, we adopt the following conventions:

• A complex space is always assumed to be reduced, paracompact and Hausdorff.

• A modification of a complex space $X$ is a proper bimeromorphic morphism $\pi \colon Y\rightarrow X$ that is locally obtained from a finite composition of blow-ups with smooth centers.

• A subnet of a net refers to a Willard subnet. More precisely, if $f\colon I\rightarrow X$ and $f'\colon I'\rightarrow X$ are nets, namely, maps from directed sets $I$ and $I'$ to the same set $X$, we say $f'$ is a subnet of $f$ if there is an increasing final function $h\colon I'\rightarrow I$ such that $f\circ h=f'$.

• A domain in $\mathbb {C}^ n$ refers to a connected open subset.

• A complex manifold is assumed to be paracompact.

• A submanifold of a complex manifold means a closed complex submanifold.

• A neighborhood is not necessarily open.

• The set $\mathbb {N}$ of natural numbers includes $0$.

• Increasing functions and decreasing functions are not necessarily strictly monotone.

• Positive forms are not necessarily strictly positive.

• The function $\mathbb {R}\ni x \mapsto x^2$ is convex instead of concave.

• We say a positive measure $\mu $ on a measurable space $X$ is carried by a measurable subset $Y$ if $\mu (X\setminus Y)=0$.

• $\mathrm{dd}^{\mathrm{c}}$ means $(2\pi )^{-1}\mathrm{i}\partial \overline{\partial }$.

Unless otherwise stated, $X$ denotes a connected compact complex manifold of dimension $n\geq 1$. A Hermitian form is a smooth real strictly positive $(1,1)$-form. The letter $\omega $ is reserved for such a form, while $\theta $ denotes an arbitrary smooth real $(1,1)$-form. Whenever $\theta $ or $\omega $ is assumed to be closed, this is stated explicitly.

We use the following normalizations:

Thus, on a disk in $\mathbb C$ with coordinate $z$, $\mathrm{dd}^{\mathrm{c}}\log |z|^2$ is the unit Dirac measure at the origin. Multiplier ideals of a quasi-plurisubharmonic function $u$ are defined with the weight $\exp (-u)$, and Monge–Ampère masses carry no factor $1/n!$.

1.1.1 Frequently used notation1A2835F01724

The following symbols are used throughout these constructions; the last column gives the place where each is defined. Notation that is used in only one chapter is introduced in that chapter.

1.1.2 Scope, attribution and conventions61B2D7362CC0

Classical potential theory is developed through spherical means, harmonic measures, Green functions, capacities, sweeping, and Martin boundary representations.

We work in real dimension $d\ge 2$, with ordinary Laplacian $\Delta $, unnormalized sphere area $s_{d-1}$, and unit ball volume $v_ d=s_{d-1}/d$. Put $\Phi (x)=|x|^{2-d}$ for $d\ge 3$ and $\Phi (x)=\log (1/|x|)$ for $d=2$. Then $-\Delta \Phi =c_ d\delta _0$, where $c_2=2\pi $ and $c_ d=(d-2)s_{d-1}$ for $d\ge 3$. Unless explicitly normalized otherwise, Green functions use this convention. Thus $G_{\rm Helms}=c_ dG_{\rm Foundations}$ and $C_{\rm Helms}=C_{\rm Foundations}/c_ d$. Superharmonic functions are lower semicontinuous, super-mean-valued and nontrivial on each component; positive means nonnegative unless strict positivity is stated. A potential is a nonnegative superharmonic function whose greatest harmonic minorant is zero. Measures are positive Radon measures unless signedness is specified. On noncompact spaces weak measure convergence in the source means vague convergence against $C_ c$.

The prerequisites are multivariable calculus, Lebesgue integration, measure decomposition, and compact-metric measure representation. The harmonic and subharmonic results used below are established in the preceding Pome sections.