Griffiths' conjecture on positivity of Chern–Weil forms

About 9 years old · traced to 2017

Let $(E,h)\to X$ be a Griffiths semipositive Hermitian holomorphic vector bundle. Let $P$ be a non-negative combination of Schur polynomials, and let $P\big(c_{\bullet}(E,h)\big)$ denote the differential form obtained by formally evaluating $P$ in the Chern forms of $(E,h)$ . Griffiths' conjecture. The differential form

P\big(c_{\bullet}(E,h)\big)

is positive. This is a classical positivity question for characteristic forms associated with Griffiths semipositive vector bundles; the source describes the statement as a modification of Griffiths' original conjecture.

References

Primary source

Filippo Fagioli, “Universal vector bundles, push-forward formulae and positivity of characteristic forms”, arXiv:2210.11157 (2022).

Additional references

5 papers in this index state this conjecture (2017–2022). The statement above is taken from the most recent of them; the others are arXiv:2210.15342, arXiv:2007.12425, arXiv:1807.03510, arXiv:1708.06713.

Posted 1 month ago · edited 2 days ago

Progress summary

Refreshed 2 days ago

Claimed solved

An unrefereed 2026 preprint claims explicit counterexamples in high rank, so the conjecture is reported false there while important low-rank cases remain established.

The conjecture asks whether every nonnegative combination of Schur forms is positive for a Griffiths-semipositive Hermitian holomorphic vector bundle. Earlier work established substantial special cases, but not the general statement.

Known results

Fagioli, 2020: for rank $3$ , $c_1(E,h)\wedge c_2(E,h)-c_3(E,h)$ is positive; $c_2(E,h)$ is positive in arbitrary rank.
Diverio et al., 2022: positivity for families of Schur-form combinations arising from universal Gysin formulas.
2026 preprint: $c_3(E,h)$ is weakly positive for Griffiths-positive bundles of rank $r\geq 3$ , with all Schur forms weakly positive in rank and dimension $3$ .

October 2026 counterexample claim

Yun-Heng Du's preprint claims compact-projective metrics that are Griffiths positive but have a non-positive top Chern form in every rank $r\geq 9$ . Since the top Chern form is among the relevant Schur forms, this would refute the conjecture in general; the claim is unverified.

Current status (as of October 2026): The conjecture remains unverified in full, but an unrefereed preprint claims to refute it for ranks $r\geq 9$ ; numerous lower-rank and special-family cases are proved.

Sources

No solutions have been posted yet.