Simpson’s nestedness conjecture
For the Simpson stratification of the Hodge moduli space of Higgs/de Rham objects over a smooth complex projective curve of genus , Simpson's nestedness conjecture asserts that the Simpson filtrations associated with strata are globally compatible with specialization: whenever one Simpson stratum lies in the closure of another, the corresponding Simpson filtrations satisfy Simpson's proposed nestedness relation. The cited source reports counterexamples to this assertion in every rank , for both the Dolbeault and de Rham moduli spaces.
References
Primary source
Additional references
- Simpson filtrations and closure relations of Hodge moduli spaces — arXiv — Zhi Hu, Pengfei Huang, Fei Yu
Progress summary
A new unrefereed preprint reports counterexamples in every rank from four upward, so the conjecture is claimed false there but has not been independently checked.
Simpson’s nestedness conjecture concerns a proposed global nestedness structure for Simpson filtrations and closure relations. The latest report gives a systematic obstruction rather than an isolated exceptional case.
October 2026 counterexamples
Zhi Hu, Pengfei Huang, and Fei Yu report counterexamples in every rank , using Heinloth-weight monotonicity to analyze closure relations. If correct, this refutes the conjecture in those ranks; the claim is based on an unrefereed preprint.
Current status (as of October 2026): The conjecture is claimed false in every rank , but the counterexamples remain unverified and the retrieved record does not settle lower-rank cases.
Solutions 0
No solutions have been posted yet.