Birational Invariants from Hodge Structures and Quantum Multiplication

TL;DR

This paper introduces Hodge atoms, new birational invariants combining Gromov-Witten invariants and Hodge theory, with applications to rationality problems and birational geometry.

Contribution

It develops the concept of Hodge atoms from F-bundles, providing a novel invariant that detects birational properties and rationality of complex varieties.

Findings

Proves a very general cubic fourfold is not rational.

Provides a new proof of Hodge number equality for birational Calabi-Yau manifolds.

Extends the framework to motivic Galois group representations.

Abstract

We introduce new invariants of smooth complex projective varieties, called Hodge atoms. Their construction combines rational Gromov-Witten invariants with classical Hodge theory and relies on the notion of an F-bundle, which is a non-archimedean version of a non-commutative Hodge structure. The Hodge atoms arise from the spectral decomposition of the F-bundle under the Euler vector field action, and behave additively under blowups, in accordance with Iritani's blowup theorem. We compute several examples and demonstrate applications to birational geometry. In particular, we prove that a very general cubic fourfold is not rational. We also obtain a new proof of the equality of Hodge numbers of birational Calabi-Yau manifolds in any dimension. Furthermore, we show that the framework naturally extends to representations of other motivic Galois groups. This enables the theory of atoms to produce new obstructions to rationality over non-algebraically closed fields of characteristic zero as well.