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arxiv: 2605.28621 · v1 · pith:FEBXUAFQnew · submitted 2026-05-27 · 🧮 math.DG · math.CV

Non-K\"ahler metrics on complex manifolds of LVMB type

Federico Thiella This is my paper

Pith reviewed 2026-06-29 10:25 UTC · model grok-4.3

classification 🧮 math.DG math.CV
keywords LVMB manifoldsnon-Kähler complex manifoldsbalanced metricsSKT metricscharacteristic classesholomorphic bundlestoric manifoldsHermitian metrics
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The pith

A formula for the characteristic class of the bundle over LVMB manifolds obstructs balanced metrics in most cases.

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

LVMB manifolds are compact complex non-Kähler manifolds that in many cases form holomorphic bundles over compact toric manifolds, with the bundle fixed by an algebro-combinatoric datum containing a simplicial fan. The paper derives an explicit formula for the characteristic class of this bundle expressed directly from the original LVMB datum. It applies the formula to study Hermitian metrics with special curvature conditions, establishing that balanced metrics are obstructed except in limited situations. Within the broader LVMB class a new example carrying a balanced metric is exhibited, while for the LVM subclass a complete characterization of those admitting SKT metrics is given.

Core claim

For the subclass of LVMB manifolds that admit a holomorphic bundle structure over a compact toric manifold determined by an algebro-combinatoric datum, the characteristic class of the bundle admits an explicit expression in terms of that datum; this expression yields obstructions to the existence of balanced metrics in most cases, permits the construction of one new balanced example inside the LVMB class, and supplies a characterization of the LVM manifolds that admit SKT metrics.

What carries the argument

The explicit formula for the characteristic class of the holomorphic bundle expressed from the LVMB datum, which controls the obstructions and characterizations for balanced and SKT metrics.

If this is right

  • Balanced metrics are obstructed for all but special members of the class.
  • At least one new LVMB manifold carries a balanced metric.
  • LVM manifolds that admit SKT metrics are completely characterized by the class formula.
  • The geometry of each such LVMB manifold is rigidly linked to the geometry of its toric base through the class formula.

Where Pith is reading between the lines

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

  • The same class formula may be usable to test existence of other Hermitian metrics such as Gauduchon or astheno-Kähler on the same manifolds.
  • The obstructions could be compared with known results on balanced metrics for other non-Kähler classes to see whether the LVMB case is typical or exceptional.
  • The new balanced example might serve as a test case for conjectures relating balanced metrics to other invariants like the Hodge numbers.

Load-bearing premise

The results rely on restricting attention to those LVMB manifolds that admit a holomorphic bundle structure over a compact toric manifold given by an algebro-combinatoric datum with a simplicial fan.

What would settle it

An explicit LVMB manifold for which an independent calculation of the bundle characteristic class differs from the formula, or for which a balanced metric exists despite the obstruction predicted by the formula.

read the original abstract

LVMB manifolds are a class of non-K\"ahler compact complex manifolds with a remarkably rich geometry: in many cases they admit a holomorphic bundle structure over a compact toric manifold. In fact, such a bundle is determined by an algebro-combinatoric datum encapsulating a simplicial fan. This is reflected in a close relationship between the geometry of an LVMB manifold and that of its toric base space. Throughout this paper, we restrict to this subclass of LVMB manifolds. We provide a formula for the characteristic class of the bundle in terms of the original LVMB datum. Subsequently, this expression is employed to address the existence of Hermitian metrics satisfying ``special'' conditions. We consider balanced and SKT metrics, showing that the former are obstructed in most cases. Moreover, within the LVMB class, we construct a new example of a manifold admitting a balanced metric. Finally, restricting ourselves to the subclass of LVM manifolds, we characterize those admitting an SKT metric.

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 restricts to the subclass of LVMB manifolds that admit a holomorphic bundle over a compact toric manifold determined by an algebro-combinatorial datum encoding a simplicial fan. It derives an explicit formula for the characteristic class of this bundle directly from the LVMB datum, then applies the formula to obtain obstructions to the existence of balanced metrics in most cases, constructs a new example within the LVMB class that admits a balanced metric, and characterizes those LVM manifolds (a subclass) that admit an SKT metric.

Significance. If the derivation of the characteristic class formula holds, the work supplies a concrete computational tool linking the combinatorial datum to metric obstructions on a geometrically rich class of non-Kähler manifolds. The new balanced example and the SKT characterization on the LVM subclass would constitute concrete advances in the study of special Hermitian metrics beyond the Kähler setting.

minor comments (3)
  1. The abstract states that balanced metrics are obstructed 'in most cases' but does not specify the precise combinatorial condition on the fan that produces the obstruction; this should be stated explicitly in the introduction or the relevant theorem.
  2. Notation for the LVMB datum and the simplicial fan is introduced without a dedicated preliminary section; a short table or diagram summarizing the correspondence between combinatorial data and geometric objects would improve readability.
  3. The new balanced example is announced but its explicit LVMB datum is not displayed in the abstract; including the datum (or at least the fan) in the introduction would allow readers to verify the construction immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No major comments were listed in the report, so we have no specific points to address point-by-point. We will incorporate any minor suggestions during the revision process.

Circularity Check

0 steps flagged

Derivation self-contained from LVMB datum to class formula and metric results

full rationale

The paper states an explicit formula for the characteristic class expressed directly in terms of the given algebro-combinatorial LVMB datum (encoding the simplicial fan), then applies that formula to derive obstructions for balanced metrics, a new balanced example, and a characterization of SKT metrics on the LVM subclass. All claims are scoped to the subclass admitting the holomorphic bundle over the toric base; no step equates a derived quantity to a fitted input by construction, renames a known result, or relies on a load-bearing self-citation whose content reduces to the present work. The central chain is therefore independent of the target conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters, invented entities, or ad-hoc axioms are visible from the abstract. The work rests on the standard domain assumption that LVMB manifolds in the chosen subclass carry holomorphic bundle structures over toric bases determined by simplicial fans, which is treated as given background.

axioms (1)
  • domain assumption LVMB manifolds admit a holomorphic bundle structure over a compact toric manifold determined by an algebro-combinatoric datum encapsulating a simplicial fan
    Explicitly invoked when the paper restricts to this subclass and derives the characteristic class from the LVMB datum.

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