Pluripolar Sets in Complex Analysis

• Pluripolar sets are subsets in complex spaces where a plurisubharmonic function attains −∞, marking 'small' sets with zero capacity.
• They play a key role in pluripotential theory by linking complex geometry, convergence sets, and classification schemes based on psh functions.
• Their study informs removable singularities, extension theorems, and the behavior of Monge–Ampère operators in analytic contexts.

A pluripolar set is a subset of a complex analytic space distinguished by its capacity to be locally or globally "charged" by plurisubharmonic (psh) functions taking the value $-\infty$. These sets are central to pluripotential theory, complex analysis in several variables, and the paper of singular analytic and geometric phenomena. Key advances in the structure, extension, and measure-theoretic properties of pluripolar sets connect their behavior to approximation theory, complex geometry, singularity theory, and functional analysis.

1. Definitions and Fundamental Properties

A subset $E \subset \mathbb{C}^n$ (or more generally, a complex manifold) is pluripolar if for every $x \in E$ there exists a neighborhood $U$ of $x$ and a nonconstant plurisubharmonic function $u$ on $U$ such that $u = -\infty$ on $E \cap U$ (Ma et al., 2012, Su et al., 22 Apr 2024). In the global sense, $E$ is globally pluripolar if there exists a psh function $u$ on all of $\mathbb{C}^n$ with $E \subset \{y : u(y) = -\infty\}$. A stronger notion, complete pluripolar set, requires $E = \{y : u(y) = -\infty\}$ for some nonconstant $u$.

On compact complex manifolds, the class of locally pluripolar sets coincides with globally pluripolar sets; that is, every locally pluripolar set is pluripolar (Vu, 2018). This equivalence, solved by Josefson, Bedford-Taylor, and Colţoiu, applies also to Kähler and more general Hermitian settings via suitable quasi-plurisubharmonic regularizations.

Pluripolar sets are "small" by several potentials: they are always $F_\sigma$ of type $G_\delta$ (every complete pluripolar set is $G_\delta$ (Ma et al., 2012)), have zero capacity, and in $\mathbb{C}^n$ possess Hausdorff dimension at most $2n-2$ (Wang, 2022).

2. Classes and Completeness via Plurisubharmonic Functions

Siciak’s program refines the classification of pluripolar sets:

• $L$-complete: There exists $u \in \mathrm{PSH}(\mathbb{C}^n)$ such that $\sup_{x}(u(x) - \log(1 + |x|)) < +\infty$ and $E = \{u = -\infty\}$.
• $H$-complete: For $w \in \mathrm{PSH}(\mathbb{C}^{n+1})$ homogeneous of degree 1, $E = \{w(x_0, x) = 0\}$ induces $L$-complete sets via $w(x_0, x) \mapsto \log w(1, x)$.
• $J$-complete: Defined via extremal polynomial approximation quantities $T_\ell(x, E \cap K)$, every closed $L$-complete set is $J$-complete (Ma et al., 2012).

These completeness notions link pluripolar sets to subclasses of psh functions with strong growth or regularity properties, making them central in capacity theory and the structure of convergence sets for formal power series.

3. Pluripolar Hulls and Their Extensions

The pluripolar hull $E^*$ of a pluripolar set $E$ is the intersection of all complete pluripolar sets containing $E$ (Chen et al., 2017, Wiegerinck, 2022). Formally,

$$E^* = \bigcap \{Z \in \mathbb{P}^n : \varphi(Z) = -\infty,\, \varphi \in PSH_\omega(\mathbb{P}^n),\, \varphi|_E \equiv -\infty\}.$$

The hull always yields an $F_\sigma$ set in projective space. In analytic contexts, the hull may be strictly larger than the original set, particularly for graphs of holomorphic functions with fine analytic continuation. For instance, finely holomorphic extensions over positive capacity Cantor-like sets result in hulls with multiple "sheets"—the hull includes all points corresponding to the maximal finely analytic extensions, even if classical holomorphic continuation fails (Wiegerinck, 2022).

4. Interaction with Pluripotential Theory and Monge–Ampère Operators

The complex Monge–Ampère operator $(dd^c u)^n$ may place mass on pluripolar sets if the psh function $u$ is singular along them. The refined non-pluripolar Monge–Ampère operator is constructed via truncation to ensure that mass is discounted on pluripolar sets, typically as

$$(dd^c u)^p_{np} := \lim_{l \to \infty} 1_{\{u > -l\}}(dd^c\max(u,-l))^p.$$

The class $\mathcal{G}(\Omega)$ (or its global counterpart $\mathcal{G}(X,\omega)$) consists of psh functions with finite non-pluripolar energy integrals

$$E_{np}(u) = \sum_{j=0}^n \int_\Omega u\,(dd^c u)^j \wedge \omega^{n-j} < +\infty.$$

The loss of mass formula connects the total mass of the Kähler form $\omega^n$ on a compact manifold $X$ to the mass of the non-pluripolar product: $$\int_X \omega^n - \int_X (dd^c u + \omega)^n = \sum_{j=1}^{n-1} \int_X S_j(u) \wedge \omega^{n-j},$$ where $S_j(u) = [dd^c u]^j - (dd^c u)^j_{np}$, each $S_j(u)$ is supported on the pluripolar set where $u$ is extremely singular (Andersson et al., 2021, Su et al., 22 Apr 2024).

5. Extension and Removability Theorems

Pluripolar sets are removable for large classes of functions and sheaves. In particular, any psh function defined on the complement of a compact complete pluripolar set $K \subset \Omega \subset \mathbb{C}^n$ ($n \geq 2$) uniquely extends as a psh function to $\Omega$ (Wang, 2022). This complements Shiffman's theorem, which applies to sets of Hausdorff $(2n-2)$-measure zero; indeed, complete pluripolar sets can have positive measure but still be removable for psh functions. Analogous extension results hold for separately holomorphic functions on generalized $(N, k)$-crosses with pluripolar singularities (Zajęcka, 2011).

6. Pluripolarity of Graphs and Exceptional Sets

Graphs of analytic, quasianalytic (Denjoy or Gonchar class), and algebroid functions often turn out to be pluripolar (Ibragimov, 2010, Imomkulov et al., 2012). For instance, if $f$ is a Denjoy quasianalytic function on the $n$-torus $T^n$, then its graph $\Gamma_f = \{(z, f(z)) : z \in T^n\}$ is pluripolar in $\mathbb{C}^{n+1}$. This is proved via rational approximation theory, interpolation using Fourier series, and construction of plurisubharmonic functions and Green’s functions that blow up (become $-\infty$) on the graph.

Pluripolarity persists for graphs of multi-valued algebroid functions of the form $w^k + a_1(z) w^{k-1} + \cdots + a_k(z) = 0$ with $a_j(z)$ in suitable quasianalytic classes (Ibragimov, 2010).

7. Convergence Sets and Pluripolar Structure

A central result in the theory of formal power series is that convergence sets of divergent series are precisely countable unions of closed complete pluripolar sets in projective space (Ma et al., 2012, Chen et al., 2017). In formulas, for a formal power series $f(x_1,\dots,x_n)$, the convergence set

$$\text{Conv}_p(f) = \{ [x] \in \mathbb{P}^{n-1} : f(x_1 t,\dots,x_n t) \text{ converges near } t=0 \}$$

is $F_\sigma$ pluripolar, and every such countable union arises as a convergence set for some divergent $f$. The converse also holds: every countable union of closed complete pluripolar sets is the convergence set for a divergent series.

On special subvarieties such as $\Gamma = \{[1:z:\psi(z)] : z \in \mathbb{C}\}$ (for transcendental entire $\psi$), a subset $E \subset \Gamma$ is a convergence set iff it is a countable union of compact projectively convex sets (Chen et al., 2017).

8. Pluripolar Sets in Complex Geometry and Extension Theory

Pluripolar sets feature in the paper of singularities of positive closed currents, notably via upper Lelong level sets. For $T$ a closed positive $(1,1)$-current on a compact Kähler manifold $X$, the set $E_{+}(T)$ of points with positive Lelong numbers is a countable union of analytic subsets, each being pluripolar. The volume of components of upper Lelong level sets $\mathcal{V}_{m-m',T,W}$ is controlled sharply by cohomology classes of non-pluripolar self-products (Su et al., 22 Apr 2024): $$\sum_{V \in \mathcal{V}_{m-m',T,W}} \nu(T,V)^{m'} \cdot \text{vol}(V) \leq \frac{1}{(m-m')!} \int_W [\alpha^{m'} - \langle (T|_W)^{m'} \rangle] \wedge \omega^{m-m'}.$$

In strictly pseudoconvex domains, the core—where bounded continuous psh functions cannot be improved—can be engineered to be non-pluripolar (with nonempty interior), answering negatively the question of whether the core must be "small" (Slodkowski, 2021).

9. Connections to Real Submanifolds and Almost Complex Geometry

Pluripolar sets are characterized by their interaction with real (totally real or generic) submanifolds in almost complex manifolds. If a generic real submanifold contains a compact set of positive Hausdorff measure, it cannot be pluripolar—since plurisubharmonic functions cannot be $-\infty$ on sufficiently "large" real sets. This is formalized using plurisuperharmonic measures and pseudoholomorphic disc techniques (Sukhov, 2018).

10. Impact and Further Directions

Pluripolar sets are negligible for the extension and uniqueness of psh functions, the behavior of Monge–Ampère measures, convergence properties of divergent series, and analytic continuation phenomena. They serve as a bridge between geometric potential theory, singularity theory, and function approximation in several complex variables.

Active research continues on:

• Fine analytic topology and its influence on pluripolar hulls (Wiegerinck, 2022);
• Boundary pluripolar sets and hulls (Djire et al., 2015);
• Comparison of local and global existence for complete pluripolar sets (Liu, 2015);
• The role of non-pluripolar Monge–Ampère products in measure theory and intersection theory (Andersson et al., 2021);
• Extension theory for psh functions and removable singularities across complete pluripolar sets (Wang, 2022).

The synthesis of analytic, topological, and algebraic techniques remains vital for advancing the understanding of pluripolar sets and their ramifications across complex analysis, geometry, and mathematical physics.