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arxiv: 2605.17105 · v1 · pith:7IUCNL2Pnew · submitted 2026-05-16 · 🧮 math.CV

Abundance of Bergman metrics with constant positive holomorphic sectional curvature

Shreedhar Bhat , Soumya Ganguly , Achinta Kumar Nandi , Ming Xiao This is my paper

Pith reviewed 2026-05-20 14:35 UTC · model grok-4.3

classification 🧮 math.CV
keywords Bergman metricholomorphic sectional curvatureReinhardt domainsFubini-Study metricBrouwer fixed point theoremconstant curvature Kähler metrics
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The pith

For any m and n at least 2, uncountably many distinct Reinhardt domains in C^n have Bergman metrics locally isometric to m times the Fubini-Study metric.

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

The paper constructs, for every pair of positive integers m and n with n at least 2, an R-parameter family of Reinhardt domains in complex n-space. On each domain in the family the Bergman metric is locally isometric to m times the Fubini-Study metric, and the domains are pairwise inequivalent under their Bergman metrics. This settles an open question that had been known only for the case m equals 1 and shows that no reasonable classification of such manifolds is possible. The same construction fails in dimension one, completing the picture for manifolds whose Bergman space separates points and whose Bergman metric has constant holomorphic sectional curvature.

Core claim

For any positive integers m and n with n greater than or equal to 2, there exists an R-parameter family of Reinhardt domains in C^n such that the Bergman metric of each domain is locally isometric to m times the Fubini-Study metric, and the domains within each family are mutually Bergman inequivalent.

What carries the argument

Reduction of the constant-curvature condition on Reinhardt domains to a continuous mapping problem on a suitable function space, to which the Brouwer fixed-point theorem is applied to produce the desired domains.

If this is right

  • Such domains exist in every dimension n at least 2 for every positive integer m.
  • The collection of complex manifolds whose Bergman metric has constant positive holomorphic sectional curvature is uncountable in each dimension n at least 2.
  • No classification up to Bergman isometry is feasible for these manifolds.
  • The study of manifolds with separating Bergman space and constant holomorphic sectional curvature Bergman metric is now complete.

Where Pith is reading between the lines

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

  • Similar fixed-point constructions may produce examples with other prescribed curvature functions or other Kähler metrics on Reinhardt domains.
  • The inequivalence of the domains suggests that the moduli space of such Bergman metrics is large and non-discrete.
  • One could test whether the same domains admit other geometric structures, such as constant scalar curvature or Einstein metrics, that are compatible with the Bergman metric.

Load-bearing premise

The curvature condition for the Bergman metric on these Reinhardt domains reduces to a continuous mapping problem on a function space where the Brouwer fixed-point theorem guarantees solutions.

What would settle it

An explicit computation, for some m and n greater than or equal to 2, of the Bergman metric on a candidate Reinhardt domain that fails to be locally isometric to m times the Fubini-Study metric, or a proof that all such domains are Bergman equivalent.

read the original abstract

An outstanding open question, which has attracted renewed attention following the pioneering work of Huang--Li--Treuer, is whether, for a given positive integer $m$, there exists a complex manifold whose Bergman metric is locally isometric to $m$ times the Fubini--Study metric. Previously, this question had only been resolved in the case $m=1$. In this paper, we construct, for any pair of positive integers $(m,n)$ with $n \geq 2$, an $\mathbb{R}$-parameter (hence uncountable) family of Reinhardt domains in $\mathbb{C}^n$ whose Bergman metrics are all locally isometric to $m$ times the Fubini--Study metric. Moreover, we show that the domains in this family are mutually Bergman inequivalent. This not only answers the folklore question, but also suggests that a reasonable classification of the geometry of such complex manifolds is infeasible. We also note such examples cannot exist in dimension one. The results complete the remaining open case in the study of complex manifolds whose Bergman space separates points and whose Bergman metric has constant holomorphic sectional curvature. Our approach differs from existing methods in the literature. We reduce the construction to a mapping problem and apply a Brouwer fixed point argument to establish the existence of the desired domains.

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 / 2 minor

Summary. The manuscript constructs, for any positive integers m and n with n ≥ 2, an ℝ-parameter family of Reinhardt domains in ℂ^n whose Bergman metrics are locally isometric to m times the Fubini-Study metric. It further establishes that the domains in this family are mutually Bergman inequivalent. The construction reduces the constant curvature condition to a mapping problem on a suitable function space, to which the Brouwer fixed-point theorem is applied. The paper notes that such examples do not exist in complex dimension one and completes the classification for cases where the Bergman space separates points and the Bergman metric has constant holomorphic sectional curvature.

Significance. If the central claims hold, this work provides an abundance of examples answering a folklore question that was previously resolved only for m=1. The existence of an uncountable family of mutually inequivalent domains suggests that a reasonable classification of such manifolds is infeasible. The approach using a reduction to a Brouwer fixed-point argument is a strength, offering a topological existence proof that differs from prior methods in the literature. This completes the study of complex manifolds with separating Bergman space and constant positive holomorphic sectional curvature in the Bergman metric.

minor comments (2)
  1. The abstract refers to an 'R-parameter family' without explicitly clarifying that this denotes a one-real-parameter family; adding this clarification would improve readability for readers outside the immediate subfield.
  2. The inequivalence argument relies on a continuous invariant of the Bergman metric; a brief explicit statement of this invariant in the introduction would strengthen the presentation of the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. The referee correctly highlights the use of the Brouwer fixed-point theorem in our construction and the implications for the infeasibility of classification. We appreciate the recognition that this work completes the study for complex manifolds with separating Bergman space and constant positive holomorphic sectional curvature.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper reduces the constant holomorphic sectional curvature condition for the Bergman metric on Reinhardt domains to a continuous mapping problem on a suitable function space, then invokes the Brouwer fixed-point theorem to establish existence of an R-parameter family of domains. This is an existence argument relying on an external topological theorem applied to a derived problem, rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation. The mutual inequivalence follows from a continuous invariant distinguishing family members. No equation or step in the described derivation chain reduces by construction to its own inputs; the construction is self-contained against the Brouwer theorem and the reduction step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof relies on the standard Brouwer fixed-point theorem once the Bergman curvature condition is translated into a mapping problem; no free parameters or new entities are introduced in the abstract description.

axioms (1)
  • standard math Brouwer fixed-point theorem applies to the continuous mapping obtained by reducing the constant holomorphic sectional curvature condition on Reinhardt domains.
    The abstract states that the construction reduces to a mapping problem to which Brouwer's theorem is applied.

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