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math.AG — Algebraic Geometry

Signed Mukai diagonals and derived equivalences of fivefolds

Contributed by Benjamin Antieau

For every Hochschild diagonal of a smooth projective complex variety, a Fourier–Mukai equivalence preserves both the ordinary Hodge-number sum and the sum signed by the parity of the antiholomorphic degree. The latter is the signature of a Hermitian form obtained from the generalized Mukai pairing. In dimension five these invariants determine $h^{0,3}$ and $h^{1,4}$ and reduce the unrestricted Hodge-number problem to two numerical parameters, one of which is the possible failure of invariance of $h^{0,2}$ . Albanese methods remove this structure-sheaf ambiguity when the Albanese image has dimension at least three. The general fivefold case remains open, while a theorem of Abuaf proves invariance of all Hodge numbers for fivefolds with trivial canonical bundle.

Primarily AI-generated textHuman understanding: no partsDerived categoriesfivefolds

Sep 30, 2026PDF ↗View work →

hexagon:2609.00104v1

math.AG — Algebraic Geometry

Derived invariance of the signature and Hodge numbers of fourfolds

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

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.

Primarily AI-generated textHuman understanding: all partsDerived categoriesFourier–Mukai equivalencefourfolds

Sep 26, 2026PDF ↗View work →

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