HEXAGONbeta

Derived invariance of the signature and Hodge numbers of fourfolds

Primarily AI-generated textHuman understanding: all partsmath.AG — Algebraic Geometry

Contributed by Benjamin Antieau ↗, Andrei Căldăraru, Ruoxi Li, Akhil Mathew, Joshua Mundinger, Noah Olander, Martin Olsson

SubmitterBenjamin Antieau

Version 1 / Sep 26, 2026 / CC BY 4.0

Abstract

A Fourier–Mukai equivalence between smooth proper complex varieties of common even dimension preserves the topological signature. The proof extracts the signature from the symmetrization of the Mukai pairing on even cohomology. Combining this observation with known derived invariants and the Hochschild–Kostant–Rosenberg decomposition shows that derived-equivalent smooth projective fourfolds over any field of characteristic zero have the same Hodge numbers. This argument and text was produced by ChatGPT 5.6 Sol.

Provenance statement

The initial prompt to ChatGPT 5.6 Sol was "There is a conjecture that if and are derived equivalent smooth projective varieties over the complex numbers, then they have the same Hodge numbers. Can you find a counterexample to this conjecture, or prove it?" After some work, the fourfold result was produced.

Tools used

OpenAI
ChatGPT 5.6 Sol

References

  1. Abuaf, Roland. Homological units. International Mathematics Research Notices, vol. 2017, no. 22, pp. 6943–6960. 2017DOI
  2. Addington, Nicolas; Bragg, Daniel. Hodge numbers are not derived invariants in positive characteristic. Mathematische Annalen, vol. 387, no. 1–2, pp. 847–878. 2023. With appendices, including one by Alexander PetrovDOI
  3. Caldararu, Andrei. The Mukai pairing, II: the Hochschild–Kostant–Rosenberg isomorphism. Advances in Mathematics, vol. 194, no. 1, pp. 34–66. 2005DOI
  4. Huybrechts, Daniel. Fourier–Mukai Transforms in Algebraic Geometry. Oxford University Press. 2006
  5. Navarro, Alberto; Navarro, Jose. On the Riemann–Roch formula without projective hypothesis. Transactions of the American Mathematical Society, vol. 374, no. 2, pp. 755–772. 2021
  6. Orlov, Dmitri. Derived categories of coherent sheaves and motives. Russian Mathematical Surveys, vol. 60, no. 6, pp. 1242–1244. 2005
  7. Voisin, Claire. Hodge Theory and Complex Algebraic Geometry I. Cambridge University Press, vol. 76. 2002

Version history

  1. v1Revised after moderator-requested changesCurrentSep 26, 2026