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

- ID: `P2920`
- Reference: `generic-kahler-laplacian-simple-spectrum`
- Page: https://theoremdb.org/statements/P2920
- Record maturity: Reviewed problem with recorded work

## 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?

### Definitions

- **Definition.** 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.** 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.

### What 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.

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

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.

- Original intake status: 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.
- The live MathOverflow page was checked on 2026-07-27. It is unclosed, has no accepted answer, and asks exactly for generic simplicity within a fixed Kahler class.
- 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: 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].

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `generic-kahler-laplacian-simple-spectrum`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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. https://mathoverflow.net/questions/32810/are-the-eigenvalues-of-the-laplacian-of-a-generic-k%C3%A4hler-metric-simple
   - Also cited at See dataset.references[0] for the exact external source and locator.
   - Also cited at Editorial research route recorded 2026-07-31
   - forum; reference source; checked 2026-07-31
   - Source use: citation_only
   - 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.
2. <a id="reference-2"></a>Victor Guillemin, Eveline Legendre, and Rosa Sena-Dias, “Simple Spectrum and Rayleigh Quotients”. Contemporary Mathematics (2014), 33-44. DOI 10.1090/conm/630/12664. Book section relevant to Simple Laplace spectrum for a generic metric in a Kahler class. https://doi.org/10.1090/conm/630/12664
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
3. <a id="reference-3"></a>Jerrold B. Tunnell, “Local ε-Factors and Characters of GL(2)”. American Journal of Mathematics 105(6) (1983), 1277. DOI 10.2307/2374441. Full journal article relevant to Simple Laplace spectrum for a generic metric in a Kahler class. https://doi.org/10.2307/2374441
   - scholarly_publication; reference source; checked 2026-08-01
   - Source use: citation_only
   - 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.
4. <a id="reference-4"></a>Mouhamed Moustapha Fall, Marco Ghimenti, Anna Maria Micheletti, and Angela Pistoia, “Generic properties of eigenvalues of the fractional Laplacian”. arXiv:2304.07335 (2023). Full preprint relevant to Simple Laplace spectrum for a generic metric in a Kahler class. https://arxiv.org/abs/2304.07335
   - preprint; reference source; arXiv:2304.07335, checked 2026-07-31; checked 2026-07-31
   - Source use: citation_only
   - 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.