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Kaledin–Lehn formality conjecture for polystable sheaves on K3 surfaces

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Let XX be a projective K3 surface with a generic polarization HH, and let FF be an HH-polystable coherent sheaf on XX. Kaledin–Lehn's formality conjecture. The differential graded algebra

RHom⁡ \circle*1.5(F,F)\operatorname{RHom}^{\:\raisebox{3pt}{\text{\circle*{1.5}}}}(F,F)

is formal. The paper's abstract states that this conjecture is proved for complex projective K3 surfaces, so the claim is resolved.

References

Primary source

Nero Budur and Ziyu Zhang, “Formality conjecture for K3 surfaces”, arXiv:1803.03974 (2019).

Additional references

2 papers in this index state this conjecture (2015–2018). The statement above is taken from the most recent of them; the others are arXiv:1505.00759.

derived-categoriesdg-algebrask3-surfaces
Posted

Progress summary

Refreshed
Claimed solved

A 2018 paper claims to prove the conjecture for every polystable sheaf on a projective K3 surface, with no publicly identified correction or refutation.

Kaledin and Lehn conjectured formality of the derived endomorphism algebra of an HH-polystable sheaf on a projective K3 surface. The conjecture is claimed to be settled by Budur and Zhang.

Known results

  • Kaledin and Lehn proved special cases of the conjecture.
  • Zhang proved additional special cases before the claimed complete resolution.
  • Budur and Zhang’s result covers arbitrary polarizations and also polystable derived objects for generic Bridgeland stability conditions.

2018 claimed proof

The paper Formality conjecture for K3 surfaces states a complete proof for every HH-polystable coherent sheaf, including arbitrary polarizations. A later 2019 paper describes this as a recent proof, while distinguishing associative DG-algebra formality from the weaker DG-Lie-algebra statement.

Current status (as of September 2026): The conjecture is claimed solved by Budur and Zhang, but this automated report records the published claim as unverified; no counterexample, gap, withdrawal, or retraction was found.

Sources

Solutions 0

No solutions have been posted yet.