For every smooth projective complex variety X and reductive algebraic group G, let h_sp:M_G-Higgs(X)→A_sp(X,G) be the spectral morphism induced by the Hitchin morphism, where A_sp(X,G) is the spectral base. The Chen–Ngô conjecture asserts that h_sp is surjective.

Chen–Ngo conjecture on spectral morphism surjectivity

Chen–Ngô conjectured that the spectral-data morphism is surjective. The conjecture was introduced by Tsao-Hsien Chen and Ngo Bao Chau in 2019 and remains unsettled outside established special cases.

Known results

$GL_r$ on smooth projective surfaces: confirmed before 2023 (He–Liu).
$GL_2$ on any projective variety: confirmed in 2023 (He–Liu).
Ruled surfaces and certain elliptic-fibration blowups: all reductive groups, 2025.
Further surface cases include odd-rank $SL_n$ , products of curves for $SL_n$ , and products of curves for $Sp_{2n}$ , 2026.

September 2026 claimed counterexamples

A report dated September 15, 2026 links a preprint by Siqi He, Jie Liu, and Siqing Zhang claiming failure for $GL_r$ in dimensions $n\ge 9$ , for $SL_2$ in every dimension $n\ge 2$ , and for semistable $GL_3$ Higgs bundles on smooth projective surfaces. It attributes the failure to local Cohen–Macaulay obstructions; the claim is unverified.

Current status (as of September 2026): Several special cases are proved, while the general conjecture is claimed to be refuted by new counterexamples whose correctness and precise scope remain unverified.

Chen–Ngo conjecture on spectral morphism surjectivity

Primary source

Additional references

Progress summary

Solutions 0