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arxiv: 2603.04588 · v3 · submitted 2026-03-04 · 🧮 math.CV · math.PR

Central Limit Theorem for Intersection Currents of Gaussian Holomorphic Sections

Bin Guo This is my paper

Pith reviewed 2026-05-15 16:35 UTC · model grok-4.3

classification 🧮 math.CV math.PR
keywords central limit theoremgaussian holomorphic sectionsintersection currentsrandom zeroscomplex manifoldswiener chaosfeynman diagramskahler geometry
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The pith

A universal central limit theorem governs both smooth and numerical statistics of intersection currents from multiple independent Gaussian holomorphic sections in any codimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Shiffman and Zelditch proved a central limit theorem in 2010 for smooth statistics of Gaussian random zeros in codimension one on compact Kähler manifolds. This paper shows the same limiting Gaussian behavior holds for arbitrary codimensions and for both smooth test functions and numerical counts arising from several independent sections. A reader would care because the result supplies explicit control on fluctuations of random zeros and intersections without restricting to the classical hypersurface case. The argument rests on transferring Wiener chaos expansions and Feynman diagram techniques to the setting of currents.

Core claim

We establish a universal CLT that holds for both types of statistics arising from several independent Gaussian sections, thereby fully extending the Shiffman--Zelditch theorem. The proof builds on a new geometric framework that lifts the probabilistic tools of Wiener chaos and Feynman diagrams from scalar processes to random currents on complex manifolds, providing a robust mechanism for analyzing fluctuations in random complex geometry beyond the classical codimension-one setting.

What carries the argument

The lifting of Wiener chaos expansions and Feynman diagrams from scalar random processes to random currents, which produces the variance formulas and Gaussian limits for intersection statistics in every codimension.

If this is right

  • Intersection currents of Gaussian sections converge in distribution to a Gaussian random variable after centering and scaling.
  • Both smooth statistics given by integration against test forms and numerical statistics given by counting zeros obey the same central limit law.
  • The limiting variance is determined by a universal expression involving the Kähler form and the number of independent sections.
  • The result applies uniformly on any compact Kähler manifold without further curvature assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same lifting technique could be used to study fluctuations of random holomorphic maps or sections of vector bundles in higher rank.
  • Numerical experiments on low-dimensional projective spaces for codimension two would provide a direct check of the predicted variance formula.
  • The framework suggests analogous limit theorems may hold for random subvarieties defined by systems of sections in non-Kähler settings.

Load-bearing premise

The Wiener chaos and Feynman diagram machinery transfers from scalar random variables to currents on manifolds while preserving its ability to compute limiting distributions.

What would settle it

On projective space of dimension two, compute the normalized intersection number of two random sections of high degree and check whether the histogram converges to a standard normal as degree tends to infinity.

read the original abstract

In 2010, Shiffman and Zelditch proved a central limit theorem (CLT) for smooth statistics of Gaussian random zeros in codimension one over compact K\"ahler manifolds. They raised the question of whether this result admits a two-fold generalization -- to arbitrary codimensions and to both smooth and numerical statistics -- which has remained open since then. In this paper we resolve this long-standing problem. We establish a universal CLT that holds for both types of statistics arising from several independent Gaussian sections, thereby fully extending the Shiffman--Zelditch theorem. The proof builds on a new geometric framework that lifts the probabilistic tools of Wiener chaos and Feynman diagrams from scalar processes to random currents on complex manifolds, providing a robust mechanism for analyzing fluctuations in random complex geometry beyond the classical codimension-one setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims to prove a universal central limit theorem for the fluctuations of intersection currents formed by several independent Gaussian holomorphic sections on compact Kähler manifolds. This extends the 2010 Shiffman-Zelditch result from codimension one to arbitrary codimensions while covering both smooth and numerical statistics, via a new framework that lifts Wiener chaos expansions and Feynman diagram rules from scalar processes to random currents.

Significance. If the lifting construction is rigorously justified, the result resolves a long-standing open question by providing a uniform CLT whose variance formulas specialize correctly to the codimension-one case and whose combinatorial factors remain controlled by the independence of the Gaussian sections. The framework supplies a parameter-free derivation of the moment asymptotics and opens the possibility of analyzing fluctuations in higher-codimension random geometry.

minor comments (3)
  1. [§2.3] §2.3, Definition 2.4: the pairing between the lifted chaos expansion and test forms is introduced without an explicit low-codimension example; adding a codimension-two illustration would clarify the notation for numerical statistics.
  2. [Theorem 4.1] The statement of the uniform error bound in Theorem 4.1 refers to 'sufficiently large N' without an explicit dependence on the manifold's geometry; a short remark on the constant's dependence on the Kähler class would improve readability.
  3. [Figure 3] Figure 3 (Feynman diagram for the fourth moment): the edge labels are slightly compressed; increasing the font size or adding a legend would enhance clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The report recognizes the resolution of the long-standing open question extending the Shiffman-Zelditch theorem to arbitrary codimensions and both smooth and numerical statistics. No specific major comments are listed in the report, so we have no point-by-point responses to provide. We will make any minor editorial adjustments in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs a new geometric framework by lifting Wiener chaos expansions and Feynman diagram rules directly from scalar processes to random currents on Kähler manifolds, deriving the necessary combinatorial factors and moment asymptotics from the underlying Kähler geometry and independence of the Gaussian sections. This lifting is self-contained and does not reduce any central claim to a prior result by definition or fitted parameter; the 2010 Shiffman-Zelditch theorem is cited only for context and the codimension-one specialization emerges as a correct special case rather than an input. No self-citation load-bearing steps, ansatz smuggling, or renaming of known results appear in the load-bearing reductions. The derivation remains independent of any fitted inputs or self-referential definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the new geometric framework that lifts Wiener chaos and Feynman diagrams to random currents; the setup assumes standard Gaussian measures on sections of line bundles over compact Kähler manifolds.

axioms (2)
  • domain assumption Holomorphic sections of a line bundle over a compact Kähler manifold carry a Gaussian probability measure induced by the L2 inner product from the Kähler metric.
    This is the standard probabilistic setup for Gaussian holomorphic sections as used in the cited 2010 result.
  • domain assumption The manifold is compact and Kähler, permitting the definition of intersection currents and the application of Hodge theory.
    Required for the geometric setting in which intersection currents are defined and analyzed.
invented entities (1)
  • Lifted Wiener chaos for random currents no independent evidence
    purpose: To analyze fluctuations of intersection currents in higher codimensions by extending scalar probabilistic tools.
    New construction introduced in the paper to handle the generalization beyond codimension one.

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