Around a class version of the Hodge index theorem for singular varieties

TL;DR

The paper advances the understanding of a characteristic-class version of the Hodge index theorem for singular varieties by clarifying the relationship between Goresky–MacPherson $L$-classes and Hodge-theoretic Hirzebruch classes, through the motivic transformation $T_{y*}$, its Hodge-theoretic counterpart $MHT_{y*}$, and the cobordism-based Pol transformation. It surveys known positive results and introduces new cases where the conjecture $L_*(X)=IT_{1*}(X)$ holds, including all compact toric varieties, Schubert varieties, Richardson and intersection varieties, simply connected spherical varieties, and all compact complex surfaces and threefolds. The work develops the structural framework via cobordism groups of self-dual complexes and mixed Hodge modules, and establishes Atiyah– Meyer type formulas that connect $MHT_{1*}$ and $L_* ext{(Pol)}$ for smooth pure Hodge modules. It further identifies and exploits degree-wise coincidences on the top non-rational homology locus, and introduces the notion of (virtually) stratum-wise constant complexes to guarantee BSY-type equalities in broad geometric contexts, including toric and Schubert varieties and group-action settings. Overall, the paper broadens the class of varieties for which the conjectured correspondence between L-classes and Hodge-theoretic Hirzebruch classes holds, providing new tools and criteria for verifying the conjecture in non-rational and singular cases.

Abstract

We give an overview of recent developments around a characteristic class version of the Hodge index theorem for singular complex algebraic varieties. This was formulated by Brasselet-Schuermann-Yokura as a conjecture expressing the Goresky-MacPherson homology L-classes in terms of suitable Hodge-theoretic L-classes. Along the way, we clarify the relationship between several notions of L-classes appearing in the literature, but we also include many new cases for which the conjecture is true, e.g., all compact toric varieties, all (matroid) Schubert varieties, all Richardson and intersection varieties, all projective simply connected spherical varieties, and all compact complex algebraic surfaces and threefolds.

Theorems & Definitions (86)

• Conjecture 1.1: Characteristic class version of Hodge index theorem
• Remark 1.2
• Example 1.3
• Example 1.4
• Example 1.5
• Remark 1.6
• Remark 1.7
• Example 1.8: Secant varieties of a smooth projective curve
• Example 1.9
• Remark 1.10