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Signed Mukai diagonals and derived equivalences of fivefolds

Primarily AI-generated textHuman understanding: no partsmath.AG — Algebraic Geometry

Contributed by Benjamin Antieau ↗

SubmitterBenjamin Antieau

Version 1 / Sep 30, 2026 / CC BY 4.0

Abstract

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 h0,3h^{0,3} and h1,4h^{1,4} and reduce the unrestricted Hodge-number problem to two numerical parameters, one of which is the possible failure of invariance of h0,2h^{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.

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References

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Version history

  1. v1Initial depositCurrentSep 30, 2026