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Torelli loci, product cycles, and the homomorphism conjecture for $\mathcal{A}_g$

This paper presents new calculations of intersection products between the Torelli locus and product loci in the moduli space of principally polarized abelian varieties using two independent geometric approaches, providing evidence that the tautological projection is a $\mathbb{Q}$ -algebra homomorphism for special cycles and constructing nontrivial elements in the Gorenstein kernels of tautological rings for certain moduli spaces of curves.

Original authors: Samir Canning, Lycka Drakengren, Jeremy Feusi, Daniel Holmes, Aitor Iribar López, Denis Nesterov, Dragos Oprea, Rahul Pandharipande, Johannes Schmitt, Zheming Sun

Published 2026-03-16

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CC BY 4.0

Original authors: Samir Canning, Lycka Drakengren, Jeremy Feusi, Daniel Holmes, Aitor Iribar López, Denis Nesterov, Dragos Oprea, Rahul Pandharipande, Johannes Schmitt, Zheming Sun

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 blueprints of a massive, infinite city called Abelian Varieties. This city is made up of complex geometric shapes (like doughnuts with many holes) that have very specific rules about how they can be stretched and twisted.

Mathematicians have been trying to map this city for decades. They have a special set of "standard blueprints" called Tautological Classes. Think of these as the basic, fundamental building blocks (like bricks, beams, and windows) that are guaranteed to exist in every part of the city. If you can describe a shape using only these standard blocks, it's considered "tautological" (or "standard").

However, the city is so huge that there are many shapes you can draw that don't look like they are made of these standard blocks. These are the "weird" or "non-standard" shapes.

The Big Question: The "Translation Machine"

The authors of this paper are investigating a magical machine called Tautological Projection (let's call it the Translator).

What it does: You feed the Translator a "weird" shape (a complex geometric cycle). It tries to translate that shape into the language of the standard blueprints. It asks: "If I ignore all the weird details and just look at the core structure, what standard shape does this look like?"
The Big Conjecture: The authors suspect this Translator is a perfect translator. They believe that if you take two shapes, translate them individually, and then multiply them, you get the exact same result as if you multiplied the shapes first and then translated the result.
- Analogy: Imagine you have two sentences in a foreign language. The conjecture says: "If I translate Sentence A and Sentence B separately, then combine them, it's the same as combining them first and then translating the whole paragraph."
- If this is true, it means the "weird" shapes aren't actually that weird; they just look complicated, but their core essence follows the same simple rules as the standard blocks.

How They Tested the Machine

To test if this Translator works, the authors looked at two specific types of shapes in the city:

The Torelli Locus (The "Curve" Shapes): These are shapes that come from the geometry of curves (like loops of string). They are very special and have a deep connection to the "Jacobian" (a way of turning a curve into a doughnut shape).
Product Cycles (The "Combo" Shapes): These are shapes formed by smashing two smaller cities together (e.g., a city of 2D doughnuts combined with a city of 3D doughnuts).

The authors asked: "If we smash a 'Curve Shape' and a 'Combo Shape' together, does the Translator handle the result correctly?"

Two Different Ways to Solve the Puzzle

The team used two completely different methods to solve this, like solving a maze by walking through it and by drawing a map from above.

Method 1: The "Excess Intersection" (The Microscope):
They looked extremely closely at where these shapes overlap. Imagine two roads crossing in a city. Sometimes they cross perfectly, but sometimes they cross in a messy, "excess" way where the geometry gets squished. They used a microscope to measure exactly how much "squish" happened and calculated the result. This required building a massive computer model of the city's streets (using trees and graphs) to count every possible way the shapes could intersect.
Method 2: The "Wall-Crossing" (The Time-Traveler):
This method is more abstract. Imagine the city has a "wall" that separates two different versions of reality. On one side, the shapes are stable and solid. On the other side, they are "unramified" (smooth and unbroken). The authors used a formula that acts like a time-travel machine, allowing them to jump from the stable side to the smooth side, do the math there (where it's easier), and then jump back to see what the answer was on the original side.

The Results

After running these complex calculations (which required supercomputers and years of work), they found:

The Translator Works! In every case they tested (up to a certain size of the city), the "perfect translation" rule held true. The weird shapes behaved exactly as if they were made of standard blocks.
New Discoveries: They found new "hidden" shapes that live in the "Gorenstein Kernel."
- Analogy: Think of the Gorenstein Kernel as a "black hole" in the city. It's a place where shapes exist, but when you try to measure them against the standard rules, they disappear (they equal zero). The authors found new ways to create these "ghost shapes" using a tool called the Abel-Jacobi Map (which is like taking a map of a curve and turning it into a point on a doughnut).

Why Does This Matter?

This paper is a huge step forward in understanding the "grammar" of these complex geometric cities.

If the conjecture is true: It means the entire city of Abelian Varieties is much more orderly than we thought. The "weird" shapes aren't chaotic; they follow a strict, simple set of rules.
The "Gorenstein Kernel": By finding these "ghost shapes," the authors are mapping out the limits of the city. They are finding the edges where the standard rules break down, which helps mathematicians understand the very boundaries of geometry.

In short: The authors built a super-advanced calculator to check if a complex geometric machine works perfectly. They tested it on the hardest possible combinations, and it passed every test. This suggests the universe of these shapes is governed by a beautiful, simple, and predictable logic.

1. Problem Statement and Context

The paper investigates the structure of the Chow ring $CH^*(A_g)$ of the moduli space of principally polarized abelian varieties ( $A_g$ ) and its relationship with the tautological ring $R^*(A_g)$ .

Tautological Ring: $R^*(A_g)$ is the $\mathbb{Q}$ -subalgebra of $CH^*(A_g)$ generated by the Chern classes of the Hodge bundle ( $\lambda_i$ ).
Tautological Projection: A canonical $\mathbb{Q}$ -linear projection operator $\text{taut}: CH^*(A_g) \to R^*(A_g)$ was constructed in previous work [9]. This operator decomposes any cycle into a tautological part and a non-tautological part.
The Homomorphism Conjecture: The central question is whether this projection is a ring homomorphism. That is, for any two cycles $\alpha, \beta \in CH^*(A_g)$ , does the equality
$\text{taut}(\alpha \cdot \beta) = \text{taut}(\alpha) \cdot \text{taut}(\beta)$
hold?
While the property holds trivially if one factor is already tautological, it is a non-trivial obstruction to being tautological when both factors are geometric cycles.

The authors focus on testing this conjecture for pairs involving:

Torelli cycles ( $[J_g]$ ): The pushforward of the fundamental class of the moduli space of compact-type curves $M_g^{ct}$ via the Torelli map.
Product cycles ( $[A_{g_1} \times \dots \times A_{g_k}]$ ): Cycles representing abelian varieties that decompose into products of lower-dimensional varieties.

2. Methodology

The authors employ two independent, complementary approaches to compute the pullback of product cycles under the Torelli map, $\text{Tor}^*([A_{g_1} \times \dots \times A_{g_k}])$ , which is the crucial step in verifying the homomorphism property.

Approach A: Excess Intersection Geometry

Fiber Product Analysis: They study the fiber product $Z = M_g^{ct} \times_{A_g} (A_{g_1} \times \dots \times A_{g_k})$ .
Stratification: Using results by Drakengren, they establish that $Z$ is a reduced Deligne-Mumford stack stratified by $g$ -colored extremal trees. These trees encode how the Jacobian of a curve decomposes into factors corresponding to the product locus.
Local Equations: They derive local equations for $Z$ near boundary strata using monomials associated with "critical paths" in the trees.
Excess Intersection Formula: They apply the excess intersection theory (specifically the residual intersection formula) to compute the pullback class as a sum of contributions from these strata, expressed in terms of tautological classes ( $\psi, \lambda, \kappa$ ) on the moduli space of curves.

Approach B: Wall-Crossing and Gromov-Witten Theory

Stable Maps: They recast the geometry in terms of families of stable maps to the universal abelian variety. Specifically, they relate the Torelli pullback to the reduced virtual fundamental class of the moduli space of stable maps $M_g^{ct}(\pi_r)$ to fibers of the universal family $\pi_r: X_r \to A_r$ .
Unramified Maps: They utilize a wall-crossing formula relating stable maps to unramified maps (maps to Fulton-MacPherson degenerations of the target).
Moduli Spaces: For $r=2$ (the case of $A_2 \times A_{g-2}$ ), the relevant moduli spaces of unramified maps are shown to be iterated blowups of moduli spaces of curves ( $M_{2,n}$ and $M_{1,n}$ ).
Computation: The wall-crossing formula expresses the desired class as a sum over "star-shaped graphs" involving evaluation morphisms and specific I-functions. This allows the computation to be reduced to tautological integrals on curve moduli spaces, implemented via the software admcycles.

3. Key Contributions and Results

A. Verification of the Homomorphism Property

The authors successfully verify the homomorphism property for new, non-trivial cases where the codimension of the intersection is within the "danger zone" (where the property is not guaranteed by dimension counting).

Theorem 7: The homomorphism property holds for the pairs:

$([J_g], [A_2 \times A_{g-2}])$ for all $2 \leq g \leq 8$ .
$([J_6], [A_3 \times A_3])$ on $A_6$ .

Note: The case $g=1$ was known from previous work. The authors expect the result to hold for all $g \geq 2$ , but computational complexity limits the verification to $g=8$ .

B. Vanishing Results

They prove that for partitions $(g_1, \dots, g_k)$ not falling into the specific low-codimension cases (i.e., where the sum of codimensions exceeds the dimension of $A_g$ minus the dimension of the tautological ring), the intersection vanishes:
$[J_g] \cdot [A_{g_1} \times \dots \times A_{g_k}] = 0$
This includes sporadic vanishings such as $[J_g] \cdot [A_4 \times A_{g-4}] = 0$ for $8 \leq g \leq 13$ .

C. New Constructions in Gorenstein Kernels

The paper introduces a new family of classes constructed via Abel-Jacobi maps on the universal fiber product $X_g^s$ .

Generalized Product Locus: They define a locus $PR_{g,s}$ on the $s$ -fold fiber product $X_g^s$ .
Tautological Projection: They compute the tautological projection of $[PR_{g,s}]$ explicitly as a determinant involving universal theta divisors ( $\theta_i$ ) and Poincaré classes ( $\eta_{ij}$ ).
Abel-Jacobi Pullback: They pull back the difference $\Delta_{g,s} = [PR_{g,s}] - \text{taut}([PR_{g,s}])$ via the Abel-Jacobi map $aj: M_{g,2s}^{ct} \to X_g^s$ .
Gorenstein Kernel: They prove that these pulled-back classes lie in the Gorenstein kernel (the kernel of the $\lambda_g$ $λ_{g}$ -pairing) of the tautological ring $R^*(M_{g,n}^{ct})$ $R^{*} (M_{g, n}^{c t})$ .
- Theorem 10': The class $aj^*(\Delta_{5,1})$ generates the 1-dimensional kernel of the $\lambda_5$ -pairing on $R^*(M_{5,2}^{ct})$ .
- They also construct non-trivial elements for $R^*(M_{4,4}^{ct})$ .

D. Conjectures

Conjecture 8: The authors propose that the tautological projection $\text{taut}: CH^*(A_g) \to R^*(A_g)$ is a $\mathbb{Q}$ -algebra homomorphism for all cycles.
Speculation 50: They conjecture that the Gorenstein kernels of $R^*(M_{g,n}^{ct})$ are generated by the classes $\Delta_{g,A}$ (Abel-Jacobi pullbacks of non-tautological product cycles) and the operations of the tautological ring (gluing, forgetting).

4. Significance

Evidence for the Homomorphism Conjecture: The paper provides the strongest evidence to date that the tautological projection is a ring homomorphism. This would imply a profound structural rigidity in the Chow ring of $A_g$ , suggesting that non-tautological cycles behave "orthogonally" to tautological ones in a multiplicative sense.
Computational Breakthrough: The successful application of the wall-crossing formula to the Torelli map for $r=2$ (product of two abelian surfaces) demonstrates the power of modern Gromov-Witten theoretic techniques in classical moduli problems.
New Generators for Kernels: The construction of explicit, geometric generators for the Gorenstein kernels of $R^*(M_{g,n}^{ct})$ resolves open questions about the structure of these rings. Previously, the existence of these kernels was known, but explicit geometric representatives were scarce.
Interplay of Geometries: The work bridges the geometry of abelian varieties, the geometry of curves (via the Torelli map), and the theory of stable maps/unramified maps, showing how wall-crossing can solve excess intersection problems that are difficult to approach via classical intersection theory alone.

In summary, the paper advances the understanding of the tautological ring of $A_g$ by verifying the homomorphism property in new cases and providing a geometric mechanism (via Abel-Jacobi maps and product cycles) to generate the "non-tautological" obstructions (Gorenstein kernels) in the tautological rings of moduli spaces of curves.