Universal compactified Jacobians: cohomological invariance and boundary combinatorics

1. Problem Statement

The paper addresses the cohomological properties of universal compactified Jacobians over the moduli space of stable curves, $\overline{\mathcal{M}}_{g,n}$.

• Context: For a fixed genus $g$, number of marked points $n$, and degree $d$, there exist families of "fine compactified Jacobians," denoted $\overline{\mathcal{J}}^d_{g,n}(\sigma)$, which depend on a choice of stability condition $\sigma$ (introduced by Pagani and Tommasi). These spaces are smooth, proper Deligne–Mumford stacks.
• The Phenomenon: While the geometry of these spaces varies significantly with the stability condition $\sigma$ and the degree $d$ (specifically, their boundary stratifications and isomorphism classes as stacks differ), a theorem by Migliorini–Shende–Viviani implies that their cohomology (with rational coefficients) is invariant.
• The Gap: The existing proof of this invariance relies on deep results regarding the Decomposition Theorem for "H-smooth" families of nodal curves. The authors seek a direct, combinatorial proof that explicitly demonstrates how the contributions from the boundary strata cancel out or rearrange to yield an invariant total cohomology, regardless of the choice of $\sigma$ or $d$.
• Secondary Questions: The paper also investigates:
  1. When are two universal Jacobians $\mathcal{J}^d_{g,n}$ and $\mathcal{J}^{d'}_{g,n}$ isomorphic as stacks over $\mathcal{M}_{g,n}$?
  2. Are their classes equal in the Grothendieck ring of varieties $K_0(\text{Var}_{\mathbb{C}})$?

2. Methodology

The authors employ a strategy of boundary stratification combined with combinatorial bijections and rigidity arguments.

A. Stratification of the Universal Stack

The authors analyze the stack $\mathcal{J}_{g,n}$ (parametrizing quasi-stable curves with admissible line bundles) by stratifying it according to the topological type of the underlying curves.

• Strata Indexing: Strata are indexed by triples $(\Gamma, \Gamma_0, \mathbf{d})$, where:
  • $\Gamma$ is a stable graph of type $(g,n)$.
  • $\Gamma_0 \subseteq \Gamma$ is a connected subgraph containing all vertices.
  • $\mathbf{d}$ is a multidegree function.
• Geometry of Strata: Each stratum is a torus bundle over a product of smaller universal Jacobians (associated with the vertices of $\Gamma$), modulo a finite automorphism group. Specifically, a stratum is isomorphic to $[\mathcal{J}_{\Gamma, \Gamma_0, \mathbf{d}} / \text{Aut}(\Gamma, \Gamma_0, \mathbf{d})]$.
• Compactified Jacobians: A Pagani–Tommasi stability condition $\sigma$ selects a specific union of these strata to form the compactified Jacobian $\overline{\mathcal{J}}^d_{g,n}(\sigma)$.

B. Reduction to Combinatorics

Using the additivity of the Hodge–Euler characteristic over stratifications (Lemma 3.4), the problem of proving cohomological invariance reduces to showing that the sum of contributions from the strata selected by $\sigma$ equals the sum for $\sigma'$.

• The authors prove that the contribution of a stratum depends only on the graph structure and the Picard torsor associated with the multidegree.
• The core technical challenge becomes constructing an equivariant bijection between the sets of multidegrees allowed by different stability conditions $\sigma$ and $\sigma'$ for a fixed graph pair $(\Gamma, \Gamma_0)$.

C. Rigidity and Hodge Structures

For the non-compactified universal Jacobians $\mathcal{J}^d_{g,n}$, the authors use Rigidity Theorems for variations of Hodge structure (Schmid, extended to stacks).

• They show that the variations of Hodge structure $R^q\pi_* \mathbb{Q}$ are independent of the degree $d$.
• This allows them to establish the isomorphism of cohomology rings for the non-compactified case, which serves as the base case for the inductive combinatorial argument.

D. Appendix Techniques (Feusi & Yin)

• Feusi: Uses the Franchetta property and continuity arguments to classify $S_n$-equivariant isomorphisms between universal Jacobians.
• Yin: Uses the Larsen–Lunts ring homomorphism $\Phi_{sb}$ from the Grothendieck ring of varieties to the monoid ring of stable birational equivalence classes. This tool detects "top" stable birational classes that cannot be cancelled, allowing a distinction between classes in $K_0(\text{Var}_{\mathbb{C}})$.

3. Key Contributions and Results

Main Theorem (Theorem 1.2)

For all $2g-2+n > 0$, all degrees $d, d' \in \mathbb{Z}$, and all Pagani–Tommasi stability conditions $\sigma, \sigma'$, there exists an isomorphism of rational Hodge structures:
$$H^*(\overline{\mathcal{J}}^d_{g,n}(\sigma); \mathbb{Q}) \cong H^*(\overline{\mathcal{J}}^{d'}_{g,n}(\sigma'); \mathbb{Q})$$
This isomorphism respects the action of the symmetric group $S_n$ (or subgroups thereof) and the mixed Hodge structure.

• Significance: This provides a direct combinatorial proof of a result previously known only via the Decomposition Theorem. It reveals a hidden symmetry in the boundary combinatorics of these moduli spaces.

Combinatorial Lemma (Lemma 3.7)

The authors prove that for any stable graph $\Gamma$ and subgraph $\Gamma_0$, there exists an $\text{Aut}(\Gamma, \Gamma_0)$-equivariant bijection between the sets of multidegrees $\mathbf{d}$ selected by any two stability conditions $\sigma$ and $\sigma'$.

• This bijection relies on the structure of the Picard group of the graph, $\text{Pic}(\Gamma)$, and the fact that the set of valid degrees forms a torsor under a finite abelian group. The proof utilizes number-theoretic arguments (Dirichlet's theorem on primes) to construct explicit bijections between torsors of different degrees.

Isomorphism Criteria (Appendix A.1 - Feusi)

The paper characterizes exactly when universal Jacobians are isomorphic as stacks:

• $\mathcal{J}^d_{g,n} \cong \mathcal{J}^{d'}_{g,n}$ over $\mathcal{M}_{g,n}$ if and only if $d \equiv d'$ or $-d \equiv d' \pmod{\gcd(2g-2, n)}$.
• This clarifies that while cohomology is invariant, the geometric objects themselves are distinct for most degrees.

Non-Invariance in the Grothendieck Ring (Appendix A.2 - Yin)

The paper demonstrates that cohomological invariance does not lift to the Grothendieck ring of varieties.

• Result: For $g \geq 22$, $[\mathcal{J}^d_g] = [\mathcal{J}^{d'}_g]$ in $K_0(\text{Var}_{\mathbb{C}})$ if and only if $d' \equiv \pm d \pmod{2g-2}$.
• Method: By analyzing the stable birational type of the Jacobians (using the fact that they are of general type for high genus), the authors show that different degrees yield non-birational varieties, hence distinct classes in the Grothendieck ring.

4. Significance and Implications

1. Combinatorial Insight: The paper demystifies the "miraculous" cohomological invariance of compactified Jacobians. It shows that the invariance arises from a precise combinatorial rearrangement of the boundary strata, rather than just a high-level topological theorem.
2. Algorithmic Computation: The stratification approach provides an algorithm to compute the Hodge–Deligne polynomials of these spaces, provided one knows the equivariant Euler characteristics of the uncompactified moduli spaces $\mathcal{J}_{g',n'}$ for lower genus/complexity.
3. Distinction of Invariants: The work highlights a sharp distinction between different types of invariants:
   • Cohomology: Invariant under stability conditions and degree shifts (within the constraints).
   • Isomorphism Class: Highly sensitive to degree and stability.
   • Grothendieck Class: Sensitive to degree (for high genus), reflecting the underlying birational geometry.
4. Future Directions: The paper opens questions regarding derived equivalence (Question A) and the invariance of Hodge numbers for higher-rank vector bundles (Question B), suggesting that while cohomology is robust, the derived category and higher-rank moduli spaces may exhibit more complex behavior.

In summary, this paper bridges the gap between the abstract geometric invariance of compactified Jacobians and the concrete combinatorics of their boundary stratifications, while simultaneously clarifying the limits of such invariance in the context of birational geometry and the Grothendieck ring.