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arxiv: 2606.05452 · v1 · pith:SUMRWJHYnew · submitted 2026-06-03 · 🧮 math.DG · math.CV

Curvature of hyperbolic complex manifolds

Kyle Broder , Herv\'e Gaussier This is my paper

Pith reviewed 2026-06-28 03:59 UTC · model grok-4.3

classification 🧮 math.DG math.CV
keywords Kähler metricsnegative bisectional curvatureKobayashi hyperbolic surfacesChern slopesholomorphic sectional curvaturehyperbolic complex manifoldsMok problemprojective surfaces
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The pith

A mechanism constructs complete Kähler metrics with negative bisectional curvature on product complex manifolds, resolving Mok's problem, and produces hyperbolic surfaces realizing any rational Chern slope in (2/7, 2/3).

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

The paper develops a mechanism to construct complete Kähler metrics possessing negative bisectional curvature. This mechanism applies to certain product complex manifolds and thereby resolves a longstanding problem attributed to N. Mok. The authors also construct projective Kobayashi hyperbolic surfaces that possess negative holomorphic sectional curvature. These surfaces realize any rational Chern slope in the open interval from 2/7 to 2/3. For slopes between 2/7 and 1/3, a Hermitian metric can have negative holomorphic sectional curvature while the Kähler-Einstein metric cannot.

Core claim

We introduce a mechanism for constructing complete Kähler metrics with negative bisectional curvature. This applies to some product complex manifolds, thereby resolving a longstanding problem attributed to N. Mok. We then construct projective Kobayashi hyperbolic surfaces with negative holomorphic sectional curvature whose Chern slopes c1²/c2 realize any s ∈ Q ∩ (2/7, 2/3). For slopes s∈ Q∩ (2/7,1/3), the corresponding surfaces admit a Hermitian metric with HSC<0, but their Kähler--Einstein metric cannot have HSC<0. We finally construct, for every s ∈ (1/2, 3), a sequence of projective Kobayashi hyperbolic surfaces that do not admit a Hermitian metric of nonpositive holomorphic sectional cur

What carries the argument

The mechanism for constructing complete Kähler metrics with negative bisectional curvature on product complex manifolds; the constructions of projective surfaces with prescribed Chern slopes and curvature properties.

If this is right

  • Some product complex manifolds admit complete Kähler metrics with negative bisectional curvature.
  • Projective Kobayashi hyperbolic surfaces with negative holomorphic sectional curvature exist for every rational Chern slope in (2/7, 2/3).
  • For rational slopes in (2/7, 1/3), the surfaces admit Hermitian metrics with negative HSC but their Kähler-Einstein metrics do not.
  • Sequences of projective Kobayashi hyperbolic surfaces without nonpositive HSC metrics exist with Chern slopes converging to any value in (1/2, 3).

Where Pith is reading between the lines

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

  • The mechanism might extend beyond product manifolds to other classes of complex manifolds.
  • The geography of realizable Chern slopes for hyperbolic surfaces with negative curvature could be denser than the stated intervals.
  • The separation between Hermitian and Kähler-Einstein metrics points to distinct curvature constraints in different metric classes.

Load-bearing premise

The introduced mechanism successfully produces the desired metrics on the specified product complex manifolds, and the constructed surfaces are Kobayashi hyperbolic with the claimed curvature and slope properties.

What would settle it

Finding a product complex manifold that admits no complete Kähler metric with negative bisectional curvature, or a rational slope in (2/7, 2/3) for which no such hyperbolic surface with negative HSC exists, would falsify the central claims.

Figures

Figures reproduced from arXiv: 2606.05452 by Herv\'e Gaussier, Kyle Broder.

Figure 1
Figure 1. Figure 1: Chern slopes of the examples obtained from Theorem 1.4 and Theo￾rem 1.5, compared with Demailly’s construction [Dem97]. References [AF21] A. Alarc´on and F. Forstneriˇc. The Calabi–Yau problem for Riemann surfaces with finite genus and countably many ends. Revista Matem´atica Iberoamericana, 37(4):1399–1412, 2021. [Aub78] T. Aubin. Equations du type Monge–Amp`ere sur les vari´et´es K¨ahl´eriennes compactes… view at source ↗
read the original abstract

The article addresses the construction and geography of negatively curved metrics on hyperbolic complex manifolds. We introduce a mechanism for constructing complete K\"ahler metrics with negative bisectional curvature. This applies to some product complex manifolds, thereby resolving a longstanding problem attributed to N. Mok. We then construct projective Kobayashi hyperbolic surfaces with negative holomorphic sectional curvature whose Chern slopes $c_1^2/c_2$ realize any $s \in \mathbf{Q} \cap \left( \frac{2}{7}, \frac{2}{3} \right)$. For slopes $s\in \mathbf{Q}\cap \left( \frac{2}{7},\frac{1}{3} \right)$, the corresponding surfaces admit a Hermitian metric with $\text{HSC}<0$, but their K\"ahler--Einstein metric cannot have $\text{HSC}<0$. We finally construct, for every $s \in \left( \frac{1}{2}, 3 \right)$, a sequence of projective Kobayashi hyperbolic surfaces that do not admit a Hermitian metric of nonpositive holomorphic sectional curvature, whose Chern slopes $c_1^2/c_2$ converge to $s$.

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 paper introduces a mechanism for constructing complete Kähler metrics with negative bisectional curvature on certain product complex manifolds, resolving a problem attributed to N. Mok. It constructs projective Kobayashi-hyperbolic surfaces with negative holomorphic sectional curvature whose Chern slopes c₁²/c₂ realize any rational s in (2/7, 2/3). For s in (2/7, 1/3), the surfaces admit a Hermitian metric with HSC < 0 but their Kähler-Einstein metric cannot; additionally, for every s in (1/2, 3), it constructs sequences of such surfaces without nonpositive HSC Hermitian metrics whose slopes converge to s.

Significance. If the constructions and verifications hold, the results would be significant for the geography of hyperbolic complex surfaces and the existence of negatively curved metrics, filling gaps in possible rational Chern slopes between known bounds and providing explicit examples that distinguish Hermitian and Kähler-Einstein curvature properties while resolving Mok's longstanding problem on product manifolds.

minor comments (2)
  1. The abstract refers to 'some product complex manifolds' without specifying which ones; a precise statement of the class of manifolds to which the mechanism applies would aid readability.
  2. Notation for Chern slopes is given as c₁²/c₂ but the manuscript should confirm consistency with standard normalization (e.g., whether c₂ is the second Chern class or its integral).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the significance of our results on resolving Mok's problem and the geography of Chern slopes for hyperbolic surfaces. No specific major comments were listed in the report, so we have no points to address point-by-point at this time. We remain available to provide additional details or clarifications on the constructions and verifications if requested.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claims consist of explicit constructions: a mechanism for complete Kähler metrics of negative bisectional curvature on certain product manifolds (resolving Mok's problem) and families of projective Kobayashi-hyperbolic surfaces realizing specified Chern slopes with negative holomorphic sectional curvature (or its absence). These are presented as new constructions rather than derivations that reduce by definition, fitted parameters renamed as predictions, or load-bearing self-citation chains. No equations or steps in the abstract or described content exhibit self-definitional reduction or imported uniqueness from prior author work. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work builds upon standard axioms in differential and complex geometry without introducing new free parameters or entities.

axioms (2)
  • standard math Existence and properties of Kähler metrics on complex manifolds
    The paper assumes standard results from Kähler geometry.
  • standard math Definition and properties of Kobayashi hyperbolicity
    Relies on established notions in complex analysis.

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