Birational Invariants from Hodge Structures and Quantum Multiplication

TL;DR

The paper introduces Hodge atoms, birational invariants derived from the spectral decomposition of non-archimedean A-model F-bundles that couple Gromov–Witten data with Hodge theory. Its main approach is to decompose the A-model F-bundle under the Euler vector field, producing atoms that behave additively under blowups and extend to motivic Galois frameworks, enabling obstructions to rationality across broader fields. The authors prove key results, including the non-rationality of very general cubic fourfolds and a new proof that birational Calabi–Yau manifolds share equal Hodge numbers, while also framing a general theory of G-atoms and motivated atoms applicable to varied motivic Galois groups. The significance lies in unifying enumerative geometry and Hodge theory to build practical birational obstructions and to open a motivic pathway for studying rationality across number fields and beyond.

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.

Figures (1)

• Figure 1: Eigenvalues of the $\mathop{\mathrm{\mathsf{Eu}}}\nolimits$-action at $b_\infty$

Theorems & Definitions (118)

• Example
• Theorem : see Theorem \ref{['thm:cubic4']}
• Theorem : see Theorem \ref{['thm:two.norms']}
• Theorem 2.3
• Remark 2.4
• Definition 2.6
• Example 2.7
• Remark 2.8
• Definition 2.9
• Example 2.10