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

This paper extends the Lefschetz (1,1)-theorem to singular varieties by proving the surjectivity of the Bloch-Gillet-Soulé cycle class map for normal surfaces with rational singularities and, more generally, by introducing an extended operational Chow group to establish surjectivity for varieties with isolated singularities.

Original authors: Ananyo Dan, Inder Kaur

Published 2026-06-03

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Ananyo Dan, Inder Kaur

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to understand the shape of a building. In the world of smooth, perfect buildings (mathematical "smooth varieties"), there is a famous rule called the Lefschetz (1,1)-theorem. It's like a master key: it tells you that every specific type of "hole" or "loop" you can find in the building's structure (a Hodge class) can be explained by a physical wall or floor you can actually build (a line bundle or a cycle).

But what happens when the building is broken? What if it has cracks, collapsed corners, or weird singularities? This is the problem the paper tackles.

Here is a simple breakdown of what the authors, Ananyo Dan and Inder Kaur, discovered:

1. The Problem: The Key Doesn't Fit the Broken Lock

When a building is broken (a singular variety), the usual rules get messy.

The Old Map: Mathematicians tried to use the standard "operational Chow group" (let's call this the Standard Blueprint) to find these physical walls.
The Failure: The authors found that for some broken buildings, the Standard Blueprint is incomplete. There are "holes" in the structure (Hodge classes) that exist mathematically, but the Standard Blueprint cannot point to a physical wall that creates them. The map is not surjective—it doesn't cover all the possibilities.

2. The Solution: The "Extended Blueprint"

To fix this, the authors invented a new tool called the Extended Operational Chow Group (let's call this the Master Blueprint).

How it works: Imagine you have a broken building. To understand it fully, you don't just look at the broken version; you imagine every possible way to repair (resolve) the building into a smooth, perfect version.
The Master Blueprint is a collection of all the valid "walls" you can find in every possible repair of the building, but with a special filter: you only keep the walls that, when you look back at the original broken building, don't create any "ghosts" or contradictions in the broken spots.
Think of it like a universal translator. The Standard Blueprint speaks only one dialect (the broken building), but the Master Blueprint speaks the language of all possible repairs and translates them back to the broken building perfectly.

3. The Big Discovery: The Master Key Works!

The authors proved two main things:

For Surface Cracks (Rational Singularities): If the building is a 2D surface with "nice" cracks (rational singularities), the old Standard Blueprint actually does work. The Lefschetz theorem holds true here.
For Isolated Potholes (Isolated Singularities): If the building has broken spots that are just single points (isolated singularities), the old Standard Blueprint fails. However, the new Master Blueprint works perfectly!
- They proved that with the Extended Operational Chow Group, every mathematical "hole" in a building with isolated cracks can be matched with a physical wall.
- In other words, they extended the Lefschetz (1,1)-theorem to cover these broken cases, but only if you use their new, bigger group.

4. Why the Old Map Failed (The "Totaro" Example)

The paper includes a specific example (by a mathematician named Totaro) to show why the old method failed.

Imagine a building where a specific curve (a line of bricks) is crushed into a single point.
In the "repair" version of this building, there is a wall that exists. But when you crush it back down to the broken version, that wall disappears or becomes "invisible" to the old Standard Blueprint.
The Master Blueprint is smart enough to remember that wall existed in the repair, even if it's invisible in the broken version, ensuring no "holes" are left unexplained.

5. The "What If" for Higher Dimensions

The paper also looks at 3D, 4D, and higher-dimensional buildings.

They aren't 100% sure if their Master Blueprint works for all dimensions yet.
However, they suggest that if a massive, unsolved puzzle called the Hodge Conjecture is true for smooth buildings, then their Extended Blueprint would definitely work for all broken buildings in any dimension.

Summary Analogy

Think of the Hodge (1,1)-classes as "missing puzzle pieces" in a picture of a broken vase.

The Old Method tries to find the pieces by looking only at the broken vase. It often fails to find all the pieces.
The New Method (Extended Group) looks at every possible way the vase could have been whole, finds the pieces there, and then carefully maps them back to the broken vase.
The authors proved that for vases with broken spots (isolated singularities), this new method finds every single missing piece, restoring the complete picture.

Technical Summary: Extended Operational Chow Group and Lefschetz (1, 1)-Theorem

Problem Statement
The classical Lefschetz (1, 1)-theorem states that for a smooth projective variety $X$ , every integral $(1, 1)$ Hodge class is the first Chern class of a line bundle. Extending this result to singular varieties presents significant challenges. For a singular variety $X$ , the cohomology group $H^2(X, \mathbb{Q})$ does not carry a pure Hodge structure; instead, it possesses a mixed Hodge structure. Consequently, the definition of $(1, 1)$ -classes must be restricted to the weight-2 graded piece, $Gr^W_2 H^2(X, \mathbb{Q}) \cap F^1 Gr^W_2 H^2(X, \mathbb{C})$ .

Previous attempts to generalize the theorem, such as checking the surjectivity of the first Chern class map from $\text{Pic}(X)$ to these classes, have failed in general (Totaro [8]). Furthermore, the Bloch-Gillet-Soulé (BGS) cycle class map, which maps the operational Chow group $A^p(X)$ to the space of $(p, p)$ -Hodge classes, is not always surjective for singular varieties, even when $p=1$ . The central problem addressed by this paper is the failure of the BGS map to be surjective for certain singular varieties and the need to identify a larger cycle group that restores surjectivity, particularly for the case of isolated singularities.

Methodology
The authors employ the framework of operational Chow groups and hyperenvelopes, building on the work of Fulton, MacPherson, and Totaro.

Operational Chow Groups: The paper utilizes the operational Chow group $A^p(X)$ , defined via the kernel of a morphism between Chow groups of smooth envelopes (hyperenvelopes) of $X$ .
Resolution of Singularities: The authors analyze the behavior of the BGS map under resolutions of singularities $\phi: \tilde{X} \to X$ . They utilize exact sequences relating $A^p(X)$ , $A^p(\tilde{X})$ , and the Chow groups of the exceptional locus $E$ .
Inverse Limits: To overcome the non-surjectivity of the standard BGS map, the authors introduce the extended operational Chow group, denoted $A^p_{\text{ext}}(X)$ . This group is defined as the inverse limit over the category of all resolutions of $X$ of a specific subset of $A^p(\tilde{X})$ . Specifically, $A^p_{\text{ext}}(\phi)$ consists of classes in $A^p(\tilde{X})$ whose cycle class maps to zero when restricted to the exceptional divisor.
Functoriality and Limits: The paper establishes that $A^p_{\text{ext}}(X)$ is the inverse limit of these subsets over the category of resolutions, ensuring that the cycle class map extends naturally to this new group.

Key Contributions and Results

Surjectivity for Rational Surface Singularities:
The authors prove that if $X$ is a normal projective surface with at worst rational singularities, the standard BGS cycle class map $cl_1: A^1(X) \to H^{1,1}_{\text{Hdg}}(X)$ is surjective. This extends the Lefschetz (1, 1)-theorem to this specific class of singular surfaces. In this case, $A^1(X) = A^1_{\text{ext}}(X)$ .
Introduction of the Extended Operational Chow Group:
The paper defines $A^p_{\text{ext}}(X)$ as the inverse limit of subsets of operational Chow groups of resolutions.
- Containment: $A^p(X) \subseteq A^p_{\text{ext}}(X)$ .
- Extension of the Map: The BGS cycle class map extends to a map $cl_p: A^p_{\text{ext}}(X) \to Gr^W_{2p} H^{2p}(X, \mathbb{Q})$ .
- Functoriality: For $p=1$ , the extended group $A^1_{\text{ext}}(X)$ is functorial with respect to morphisms between varieties with isolated singularities.
Lefschetz (1, 1)-Theorem for Isolated Singularities:
The primary result is that if $X$ has at worst isolated singularities (not necessarily a surface), the extended cycle class map $cl_1: A^1_{\text{ext}}(X) \to H^{1,1}_{\text{Hdg}}(X)$ is surjective. This demonstrates that every Hodge $(1, 1)$ -class in $H^2(X, \mathbb{Q})$ arises from an element of the extended operational Chow group, filling the gap left by the non-surjectivity of the standard map.
Counterexamples and Non-Surjectivity:
The paper provides explicit examples (following Totaro) where the standard BGS map $cl_1: A^1(X) \to H^{1,1}_{\text{Hdg}}(X)$ is not surjective. Specifically, for a surface $X$ obtained by contracting a curve $C'$ (the strict transform of a cubic) in a blow-up of $\mathbb{P}^2$ at 10 general points, $A^1(X) \neq A^1_{\text{ext}}(X)$ , and the standard map fails to be surjective. This necessitates the introduction of the extended group.
Higher Codimension and the Hodge Conjecture:
For higher codimensions ( $p > 1$ ), the surjectivity of the extended map is not guaranteed without further assumptions. The authors define a further enlargement, $A^p_{\text{wide}}(X)$ , based on a subcategory of resolutions where pullbacks of homologically trivial cycles are surjective. They show that if the Hodge conjecture holds for smooth projective varieties, then the map $cl_p: A^p_{\text{wide}}(X) \to H^{p,p}_{\text{Hdg}}(X)$ is surjective for all $p > 0$ .

Significance
The paper claims to generalize the classical Lefschetz (1, 1)-theorem to the setting of singular varieties, specifically addressing the limitations of previous approaches involving the first Chern class map and the standard operational Chow group. By introducing the extended operational Chow group, the authors provide a natural framework where the cycle class map becomes surjective for varieties with isolated singularities. This result clarifies the relationship between Hodge classes and algebraic cycles in the presence of singularities, showing that while standard cycles may not suffice, a limit over resolutions (the extended group) captures all $(1, 1)$ -classes. The work also contextualizes these findings against Arapura's results on motivic cohomology, noting that while Arapura's map is surjective onto a smaller subspace ( $F^1 H^2 \cap H^2$ ), the extended operational Chow group captures the larger space of Hodge $(1, 1)$ -classes inherent to the singular structure.