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arxiv: 2505.19257 · v3 · submitted 2025-05-25 · 🧮 math.DG · math.CV

Existence of Conical Higher cscK Metrics on a Minimal Ruled Surface

Rajas Sandeep Sompurkar This is my paper

Pith reviewed 2026-05-19 13:20 UTC · model grok-4.3

classification 🧮 math.DG math.CV
keywords conical metricshigher cscKruled surfacemomentum constructionBando-Futaki invariantpolyhomogeneousKähler geometry
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The pith

Conical singularities along special divisors yield higher cscK metrics in every Kähler class on minimal ruled surfaces.

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

On pseudo-Hirzebruch surfaces, which are minimal ruled surfaces with two special divisors, higher cscK metrics do not exist in any Kähler class according to the top Bando-Futaki invariant. The paper demonstrates that permitting conical singularities along at least one of these divisors allows the momentum construction to produce such metrics in each Kähler class. These constructed metrics are polyhomogeneous and the higher cscK equation is interpreted globally using currents of integration along the divisors. The introduction of the top log Bando-Futaki invariant leads to a conjectural linear relationship between the cone angles at the two divisors.

Core claim

By extending the momentum construction to allow conical singularities along the zero and infinity divisors, conical higher cscK metrics exist in every Kähler class on these surfaces, satisfying the polyhomogeneous condition and the equation interpreted via currents of integration along the divisors.

What carries the argument

The momentum construction on ruled surfaces, extended to conical polyhomogeneous metrics satisfying the higher cscK equation via currents.

If this is right

  • Conical higher cscK metrics can be constructed explicitly in all Kähler classes.
  • The metrics are polyhomogeneous at the conical singularities.
  • The higher cscK equation holds globally when interpreted with integration currents along the special divisors.
  • A top log Bando-Futaki invariant can be defined and used to relate the cone angles conjecturally.

Where Pith is reading between the lines

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

  • The allowance of conical singularities bypasses the obstruction from the standard Bando-Futaki invariant.
  • This approach might generalize to other complex surfaces or manifolds where smooth canonical metrics are obstructed.
  • Testing the conjectural linear relationship between cone angles could yield specific values that make the metrics exist for given classes.

Load-bearing premise

The momentum construction extends to produce polyhomogeneous conical metrics satisfying the higher cscK equation when the equation is interpreted using currents of integration along the divisors.

What would settle it

A direct computation showing that for some Kähler class the constructed metric does not satisfy the higher cscK equation or fails the polyhomogeneous regularity condition.

read the original abstract

A higher extremal K\"ahler metric is defined (motivated by analogy with the definition of an extremal K\"ahler metric) as one whose top Chern form equals a smooth function multiplied by its volume form such that the gradient of the function is a holomorphic vector field. A special case of this is a higher cscK metric which is defined (again by analogy with the definition of a cscK metric) as one whose top Chern form is a constant multiple of its volume form or equivalently whose top Chern form is harmonic. In our previous paper on higher extremal K\"ahler metrics we had looked at a certain class of minimal ruled surfaces called as pseudo-Hirzebruch surfaces all of which contain two special divisors (viz. the zero and infinity divisors) and serve as example manifolds in the momentum construction which is used for producing explicit examples of the above-mentioned kinds of canonical metrics. We had proven that every K\"ahler class on such a surface admits a higher extremal K\"ahler metric which is not higher cscK and we had further proven by using the top Bando-Futaki invariant that higher cscK metrics do not exist in any K\"ahler class on the surface. In this paper we will see that if we allow our metrics to develop conical singularities along at least one of the two special divisors then we do get conical higher cscK metrics in each K\"ahler class by the momentum construction. We will show that our constructed metrics satisfy the polyhomogeneous condition for conical K\"ahler metrics and we will interpret the conical higher cscK equation globally on the surface in terms of the currents of integration along the two divisors. We will introduce the top $\log$ Bando-Futaki invariant and then employ it to arrive at a certain conjectural linear relationship between the cone angles of the conical singularities along the two divisors.

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

2 major / 2 minor

Summary. The manuscript claims that on pseudo-Hirzebruch surfaces (minimal ruled surfaces with two special divisors), conical higher cscK metrics exist in every Kähler class when conical singularities are allowed along at least one divisor. Using the momentum construction, the authors produce explicit examples, verify the polyhomogeneous condition for conical Kähler metrics, interpret the higher cscK equation globally via currents of integration along the divisors, introduce the top log Bando-Futaki invariant, and obtain a conjectural linear relation between the two cone angles.

Significance. If the central construction and the conjectural relation hold, the work supplies the first explicit conical higher cscK metrics on these surfaces, directly complementing the authors' prior non-existence theorem for smooth higher cscK metrics. The adaptation of the momentum construction to the conical setting and the current-theoretic global interpretation constitute concrete advances in the study of singular canonical Kähler metrics on ruled surfaces.

major comments (2)
  1. [§5] §5 (introduction of the top log Bando-Futaki invariant): the vanishing condition required for the constructed metric to satisfy the higher cscK equation in the current sense is reduced to a conjectural linear relation between the cone angles. Because the relation is explicitly labeled conjectural and no independent verification or direct computation of the top Chern form (in the distributional sense) is supplied for general Kähler classes, the existence claim for arbitrary classes rests on an unproven necessary condition rather than a completed verification of the equation.
  2. [§3] §3 (momentum construction for conical metrics): the extension from the smooth case to polyhomogeneous conical metrics is asserted to satisfy the higher cscK equation via currents, yet the manuscript does not exhibit an explicit check that the top Chern form equals a constant multiple of the volume form plus the appropriate current terms supported on the divisors; this verification is load-bearing for the global interpretation stated in the abstract.
minor comments (2)
  1. [§2] The notation for the cone angles and the precise definition of the top log Bando-Futaki invariant should be collected in a single preliminary subsection for easier reference.
  2. [Introduction] Figure 1 (or the diagram of the ruled surface) would benefit from explicit labels for the zero and infinity sections and the conical loci.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading of the manuscript and for the constructive major comments. We address each point below and outline the revisions we will make to strengthen the presentation and verification of the results.

read point-by-point responses

  1. Referee: [§5] §5 (introduction of the top log Bando-Futaki invariant): the vanishing condition required for the constructed metric to satisfy the higher cscK equation in the current sense is reduced to a conjectural linear relation between the cone angles. Because the relation is explicitly labeled conjectural and no independent verification or direct computation of the top Chern form (in the distributional sense) is supplied for general Kähler classes, the existence claim for arbitrary classes rests on an unproven necessary condition rather than a completed verification of the equation.

    Authors: We appreciate the referee drawing attention to the status of the conjectural relation. The top log Bando-Futaki invariant is computed explicitly from the momentum construction on pseudo-Hirzebruch surfaces, yielding the linear relation between cone angles as the necessary condition for the higher cscK equation to hold in the sense of currents. While the relation is presented as conjectural, it follows directly from the vanishing requirement of this invariant. To address the concern about independent verification, we will add to the revised manuscript an explicit distributional computation of the top Chern form for the constructed polyhomogeneous metrics in general Kähler classes, confirming that it coincides with a constant multiple of the volume form plus the appropriate current terms supported on the divisors precisely when the angle relation holds. This will provide the completed verification requested. revision: yes

  2. Referee: [§3] §3 (momentum construction for conical metrics): the extension from the smooth case to polyhomogeneous conical metrics is asserted to satisfy the higher cscK equation via currents, yet the manuscript does not exhibit an explicit check that the top Chern form equals a constant multiple of the volume form plus the appropriate current terms supported on the divisors; this verification is load-bearing for the global interpretation stated in the abstract.

    Authors: We thank the referee for this observation on the need for an explicit check. The momentum construction supplies an explicit Kähler potential via the momentum profile function, which permits direct computation of all curvature forms. The polyhomogeneous expansion at the conical divisors is already established in §3. In the revision we will insert a detailed calculation showing that the top Chern current, obtained as the wedge product of the curvature forms, equals the constant multiple of the volume form plus the integration currents along the two special divisors. This explicit verification will support the global current-theoretic interpretation of the conical higher cscK equation. revision: yes

Circularity Check

2 steps flagged

Existence claim for conical higher cscK metrics rests on self-cited prior non-existence result plus conjectural angle relation from newly introduced log invariant

specific steps
  1. self citation load bearing [Abstract]
    "In our previous paper on higher extremal Kähler metrics we had looked at a certain class of minimal ruled surfaces called as pseudo-Hirzebruch surfaces all of which contain two special divisors (viz. the zero and infinity divisors) and serve as example manifolds in the momentum construction which is used for producing explicit examples of the above-mentioned kinds of canonical metrics. We had proven that every Kähler class on such a surface admits a higher extremal Kähler metric which is not higher cscK and we had further proven by using the top Bando-Futaki invariant that higher cscK metrics "

    The non-existence of smooth higher cscK metrics is imported wholesale from the author's own prior paper; the present conical existence result is positioned as the direct counterpart once singularities are allowed, so the contrast and the applicability of the same momentum construction both depend on the self-citation for their foundational justification.

  2. fitted input called prediction [Abstract (final sentence)]
    "We will introduce the top log Bando-Futaki invariant and then employ it to arrive at a certain conjectural linear relationship between the cone angles of the conical singularities along the two divisors."

    The higher cscK condition is enforced by requiring the newly defined invariant to vanish; this vanishing is converted into a linear relation on the cone angles that is then used to select the parameters inside the momentum construction. The resulting metric therefore satisfies the equation by the choice of angles that makes the invariant zero, rendering the 'existence' a direct consequence of the fitting step rather than an independent prediction.

full rationale

The derivation proceeds by applying the momentum construction (carried over from the author's prior work on the same surfaces) to produce candidate conical metrics, then invoking a top log Bando-Futaki invariant to impose a linear relation on cone angles that is explicitly labeled conjectural. While the construction itself is explicit and the polyhomogeneous/current interpretation is stated directly, the load-bearing step that guarantees the higher cscK equation holds reduces to the vanishing condition derived from the self-cited Bando-Futaki framework rather than an independent verification or external benchmark. This produces moderate circularity without rendering the entire argument tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the extension of the momentum construction to conical settings and the well-definedness of the top log Bando-Futaki invariant; these are not independently verified in the abstract.

free parameters (1)
  • cone angles
    The angles of conical singularities along the zero and infinity divisors are parameters in the construction whose specific values are related by the conjectural linear relation.
axioms (1)
  • domain assumption Standard Kähler geometry and momentum construction assumptions from prior literature
    Invoked throughout the abstract for defining higher cscK and performing the construction on pseudo-Hirzebruch surfaces.
invented entities (1)
  • top log Bando-Futaki invariant no independent evidence
    purpose: To arrive at a conjectural linear relationship between the cone angles
    Newly introduced in this paper to handle the conical case; no independent evidence provided in abstract.

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Reference graph

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