On the invariance of irregular Hodge numbers under crepant birational equivalences

Abstract: The Batyrev--Kontsevich theorem asserts that birational Calabi--Yau varieties have the same Hodge numbers. In this article, we prove an analogue for Landau--Ginzburg models $(U,f)$, consisting of smooth quasi-projective complex varieties $U$ with regular functions $f$ on $U$. We show that the irregular Hodge numbers of the twisted de Rham cohomology $\mathrm{H}^k_{\mathrm{dR}}(U,f)$ are invariant under crepant birational equivalences.

• The paper demonstrates that irregular Hodge numbers are preserved under crepant birational equivalences for pairs with non-degenerate functions.
• It employs geometric motivic integration and exponential mixed Hodge structures to rigorously prove the invariance result.
• The findings extend mirror symmetry principles from Calabi–Yau varieties to broader Landau–Ginzburg models, offering new computational tools.

Invariance of Irregular Hodge Numbers under Crepant Birational Equivalences

Introduction and Motivation

The paper "On the invariance of irregular Hodge numbers under crepant birational equivalences" (2607.12531) addresses a fundamental question in algebraic geometry and mirror symmetry: to what extent are invariants arising from Hodge theory preserved under birational transformations, particularly in the context of Landau–Ginzburg models. While classical Hodge numbers and Betti numbers are known to be birationally invariant among Calabi–Yau varieties, the extension to the so-called irregular Hodge numbers—especially for non-Calabi–Yau and Landau–Ginzburg models—remained open, with significant implications for mirror symmetry and the classification of algebraic varieties.

Irregular Hodge Theory and Landau–Ginzburg Models

Irregular Hodge theory, initiated by Deligne [Del07] and further developed in works such as [Sab10], [Yu14], and [ESY17], generalizes classical Hodge theory by considering twisted de Rham cohomologies associated to pairs $(U, f)$, where $U$ is a smooth quasi-projective variety and $f$ is a regular function. The paper defines the irregular Hodge numbers as the dimensions of certain graded pieces in the irregular Hodge filtration, constructed in analogy with classical Hodge numbers but adapted to the exponential mixed Hodge structure setting.

The paper restricts to pairs $(X, D, f)$, where $X$ is smooth projective, $D$ is a normal crossing divisor, and $f$ is a regular function with a non-degeneracy condition in the sense of Katz [Kat90] and Mochizuki [Moc15]. This setup generalizes mirror symmetry partners of Fano varieties, where the mirror is conjecturally a Landau–Ginzburg model $(U, f)$.

Crepant Birational Equivalences: Definitions and Geometry

Two pairs $(X, D_X, f)$ and $(Y, D_Y, g)$ with non-degenerate functions are defined as crepant if there exist birational morphisms $U$0 and $U$1 from a common smooth projective variety $U$2, such that pullbacks of the log canonical divisors agree ($U$3) and the functions $U$4 and $U$5 coincide on a Zariski-dense open subset. This notion is an adaptation of classical crepant birational equivalence, but specialized to pairs with regular functions, critically maintaining the structure needed for Hodge-theoretic invariants.

Examples include mutations of Laurent polynomials, as well as standard toric compactifications.

Main Results and Proof Strategy

Strong Claim

The central theorem is: For crepant pairs with non-degenerate functions, the associated Landau–Ginzburg models $U$6 and $U$7 have identical irregular Hodge numbers.

This extends the classical Batyrev–Kontsevich birational invariance of Hodge numbers from the Calabi–Yau to Landau–Ginzburg context, specifically incorporating irregular Hodge numbers.

Proof Outline

The proof relies on geometric motivic integration ([CL16], [Bil23]), establishing that the class of $U$8 and $U$9 coincide in the exponential Grothendieck group ExpM$f$0, from which equality of irregular Hodge numbers follows due to purity of associated exponential mixed Hodge structures. The irregular Hodge polynomial acts as a motivic measure on this group, ensuring that numerical invariants translate directly from motivic identity.

Key technical steps include:

• Reduction to cases where $f$1 and $f$2 extend to morphisms to $f$3,
• Crepant blow-ups to resolve intersection data, maintaining non-degeneracy,
• Use of the motivic change-of-variable formula in the context of arc spaces,
• Application of purity (Proposition 3.9), ensuring recovery of numerical invariants.

Where the functions' zero loci and pole divisors intersect, iterative blow-ups are performed, exploiting the independence of irregular Hodge numbers from specific non-degenerate function choices, as established in [QZ26].

Numerical Results and Contradictory Cases

The paper shows that in the strongly non-degenerate—affine case, twisted de Rham cohomology vanishes except in top degree, simplifying computations. It further demonstrates that not all crepant morphisms of pairs can be upgraded to crepant equivalences of pairs with non-degenerate functions (Proposition 2.11), illustrating the necessity of function-level data, not just geometric equivalence.

Practical and Theoretical Implications

Practically, this result assures that irregular Hodge numbers, which encode refined topological and Hodge-theoretic data relevant to mirror symmetry, are stable under crepant birational equivalences for a wide class of Landau–Ginzburg models. Theoretically, it supports the extension of mirror symmetry correspondences beyond Calabi–Yau varieties and underpins the conjectural equivalence of various candidate Hodge numbers in the Landau–Ginzburg setting, as proposed in [KKP17].

Additionally, it provides tools for computing irregular Hodge numbers via motivic integration and arc spaces, linking algebraic, analytic, and motivic methods. The interplay of exponential mixed Hodge structures, motivic integration, and birational geometry—especially their compatibility with sophisticated filtrations and cohomology theories—may inform future developments in nonabelian Hodge theory, wall-crossing phenomena, and enumerative invariants.

Speculation on Future Developments

The methods and results suggest deeper invariance phenomena for other irregular or exponential-type invariants under birational transformations. One anticipates further extensions to cases with singularities, connections to birational invariants of D-modules, and applications to the study of arithmetic and enumerative properties of mirrors. The motivic framework adopted here may be extended to generalized cohomological theories, incorporating more categorical and derived structures (e.g., in the setting of infinity categories or noncommutative geometry).

Conclusion

This paper rigorously establishes the invariance of irregular Hodge numbers under crepant birational equivalences of pairs with non-degenerate functions, thereby generalizing classical Hodge-theoretic invariance to the Landau–Ginzburg context. The approach, grounded in motivic integration and exponential mixed Hodge structures, resolves key conjectures in mirror symmetry and sets a foundation for deeper explorations of Hodge-theoretic invariants in birational algebraic geometry.