Extended operational Chow group and Lefschetz (1,1)-theorem

2025/06/16 by Ananyo Dan, Inder Kaur, Dan, Ananyo +1 · 1 citation
Mathematics · #14C15 #14C30 #32G20 #32S35 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2506.13220

openalex publication_date 2025/06/16 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

Abstract

Let X be a singular, projective variety. For every p>0, H^2p(X,ℚ) is equipped with a mixed Hodge structure. The elements of Gr^W_2pH^2p(X,ℚ) ∩ H^p,p Gr^W_2pH^2p(X,ℂ) will be called Hodge (p,p)-classes. The purpose of this article, is to study the Bloch-Gillet-Soulé (BGS) cycle class map from the p-th operational Chow group A^p(X) to the space of (p,p)-Hodge classes. We show that if p=1 and X is a normal surface with at worst rational singularities, then the BGS cycle class map is surjective. This extends the Lefschetz (1,1)-theorem to the setup of rational surface singularities. However, the BGS map is not always surjective. For this reason we introduce extended operational Chow group A^p_ext(X) which contains the operational Chow group. We show that the BGS cycle class map extends to A^p_ext(X). Moreover, if p=1 and X has at worst isolated singularity (not necessarily a surface), then the extended BGS map is surjective. This further extends the Lefschetz (1,1)-theorem to the case of isolated singularities.

Citations

• A Lefschetz (1,1) theorem for singular varieties
• Descent, Motives and K-theory

Cited by

• Hodge Conjecture via Singular Varieties

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