[#P2920] Simple Laplace spectrum for a generic metric in a Kahler class

A flat mathematical diagram showing a curved surface beside separated Laplace eigenvalue levels. — A schematic view of a curved surface beside separated Laplace eigenvalue levels.

Problem. Let \((M,J)\) be a compact connected Kahler manifold and fix a Kahler class \([\omega]\). In the \(C^\infty\) space of Kahler metrics whose Kahler forms represent \([\omega]\), is the set of metrics for which every eigenvalue of the scalar Laplace-Beltrami operator is simple a residual set?

1Context

First-variation matrices for repeated eigenvalues, transversality lemmas, and verified perturbing potentials can be reused one eigenspace at a time. The fixed-class constraint isolates the gap left by unrestricted metric perturbations.

2Problem setup

Definition 1 (A subset). A subset is residual if it contains a countable intersection of open dense subsets in the C-infinity topology on smooth Kahler potentials representing the fixed class.

Definition 2 (An eigenvalue). An eigenvalue is simple when its eigenspace on smooth complex-valued functions has dimension one; eigenvalues are counted with multiplicity and the zero eigenvalue is simple because M is connected.

Remark 1. First-variation matrices for repeated eigenvalues, transversality lemmas, and verified perturbing potentials can be reused one eigenspace at a time. The fixed-class constraint isolates the gap left by unrestricted metric perturbations.

3What counts as a solution

Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual.
A positive proof must specify the topology and establish density for simultaneous simplicity of the full countable spectrum, rather than splitting only a fixed finite collection of eigenvalues.

1Status

Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample. Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual.^[1]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 UNKNOWN as of 2026-07-31. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample. Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual.^[1] Relevance to this problem Defines the dated frontier and the exact remainder that research on this problem must resolve. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details UNKNOWN as of 2026-07-31. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample. A complete resolution must satisfy this condition: Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual. Open full record →
	Trace	Reported
attempt · Route 1 Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual.^[1] Relevance to this problem For Simple Laplace spectrum for a generic metric in a Kahler class, record generic-kahler-laplacian-simple-spectrum-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove that for every compact connected Kahler manifold and every Kahler class, the metrics with simple scalar Laplace spectrum form a residual subset of the fixed-class metric space, or exhibit a specific manifold and class for which that subset is not residual. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
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Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-31. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.

Guillemin, Legendre, and Sena-Dias, Simple Spectrum and Rayleigh Quotients, DOI 10.1090/conm/630/12664, calculate eigenvalue variation under Kahler deformations and explicitly relate the calculation to this question. The paper does not announce a general resolution.
Uhlenbeck's generic-simplicity theorem for unrestricted Riemannian metrics and later generic-spectrum results for fractional Laplacians and metric graphs were checked. Their allowed perturbations leave the fixed Kahler class and do not settle this restricted problem.
A 2026-07-27 exact-title and citation search found no later primary source claiming the general theorem or a counterexample. A local corpus search found no duplicate target.

Recorded example 1. Uhlenbeck's theorem permits arbitrary Riemannian perturbations. A perturbation used here must remain Kahler for the fixed complex structure and must keep the cohomology class [omega].

How the 2 records connectTyped relations and evidence flow

ProblemSimple Laplace spectrum for a generic metric in a Kahler class

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Hot spots conjecture for convex planar domainsspectral geometry
Planar drums whose spectra differ only finitelyspectral geometry
Hodge conjecturecomplex geometry

How to cite

TheoremDB contributors, “Simple Laplace spectrum for a generic metric in a Kahler class,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/generic-kahler-laplacian-simple-spectrum

This page as plain text: generic-kahler-laplacian-simple-spectrum.md

This problem includes 2 records joined by 1 typed links, sourced from mathoverflow.net^[1], current as of July 31, 2026.

1References

Packet source. MathOverflow: Are the eigenvalues of the Laplacian of a generic Kahler metric simple?. Question 32810 was checked on 2026-07-27. The live unclosed page has no answers or comments, and the statement here follows its fixed-complex-manifold, fixed-Kahler-class scope. Question 32810 was checked on 2026-07-27. The live unclosed page has no answers or comments, and the statement here follows its fixed-complex-manifold, fixed-Kahler-class scope. ↗forum · discovery source · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Simple Laplace spectrum for a generic metric in a Kahler class: UNKNOWN as of 2026-07-27. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.Source named by the research packet.
Victor Guillemin, Eveline Legendre, and Rosa Sena-Dias, “Simple Spectrum and Rayleigh Quotients”. Contemporary Mathematics (2014), 33-44. DOI 10.1090/conm/630/12664. Status evidence identified in the source record and checked at the linked publication. ↗journal article · primary source · checked 2026-08-01Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.Also cited at Book section relevant to Simple Laplace spectrum for a generic metric in a Kahler class.Source used to assess the problem's recorded status.For Simple Laplace spectrum for a generic metric in a Kahler class: UNKNOWN as of 2026-07-27. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.
Jerrold B. Tunnell, “Local ε-Factors and Characters of GL(2)”. American Journal of Mathematics 105(6) (1983), 1277. DOI 10.2307/2374441. Status evidence identified in the source record and checked at the linked publication. ↗journal article · primary source · checked 2026-08-01Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.Also cited at Full journal article relevant to Simple Laplace spectrum for a generic metric in a Kahler class.Source used to assess the problem's recorded status.For Simple Laplace spectrum for a generic metric in a Kahler class: UNKNOWN as of 2026-07-27. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.
Mouhamed Moustapha Fall, Marco Ghimenti, Anna Maria Micheletti, and Angela Pistoia, “Generic properties of eigenvalues of the fractional Laplacian”. arXiv:2304.07335 (2023). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:2304.07335, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.Also cited at Full preprint relevant to Simple Laplace spectrum for a generic metric in a Kahler class.Source used to assess the problem's recorded status.For Simple Laplace spectrum for a generic metric in a Kahler class: UNKNOWN as of 2026-07-27. The MathOverflow page remains unanswered. Guillemin, Legendre, and Sena-Dias derive the relevant first-variation formulas in a paper devoted to this exact question, while the dated citation and exact-phrase search found no later proof or counterexample.

An original CC0 textbook restatement motivated by the cited MathOverflow question; the topology and multiplicity convention are stated explicitly.

[#P2920] Simple Laplace spectrum for a generic metric in a Kahler class

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance