Universal compactified Jacobians: cohomological invariance and boundary combinatorics

Abstract: Pagani and Tommasi have introduced a class of smoothable fine compactified Jacobians $$\overline{\mathcal{J}}<em>{g,n}^{d(σ)\rightarrow} \overline{\mathcal{M}}</em>{g,n}$$ over the moduli space of stable curves, depending nontrivially on the degree $d$ and the choice of a stability condition $σ$. A theorem of Migliorini-Shende-Viviani implies that the cohomology of $\overline{\mathcal{J}}<em>{g,n}<sup>d(σ)$ is independent of $d$ and $σ$, a statement which is quite unexpected from the point of view of the boundary geometry of these spaces. We reprove this independence statement using a direct combinatorial argument, summing up contributions of individual strata. The Appendix includes a result by J. Feusi characterizing when $\mathcal{J}</em>{g,n}<sup>d$ and $\mathcal{J}<em>{g,n}<sup>{d&#39;}$ are $S_n$-equivariantly isomorphic over $\mathcal{M}</em>{g,n}$, and a result by Q. Yin showing that $[\mathcal{J}<sup>d_g]$ and $[\mathcal{J}<sup>{d&#39;}_g]$ are not always equal in $K_0(\text{Var}_{\mathbb{C}})$.

• The paper proves that the cohomology groups, including Hodge numbers and orbifold Euler characteristics, are invariant under variations in degree and combinatorial stability conditions.
• It employs a direct combinatorial proof that pairs contributions from individual boundary strata to explicitly match Hodge contributions across different moduli spaces.
• It highlights that while additive cohomology is invariant, the cup-product and ring structure vary with stability, necessitating perverse filtered intrinsic invariants.

Universal Compactified Jacobians: Cohomological Invariance and Boundary Combinatorics

Introduction and Background

The study of universal compactified Jacobians over the moduli space of stable curves $\overline{\mathcal{M}}_{g,n}$ forms a keystone of modern algebraic geometry. This paper explores the class of smoothable fine compactified Jacobians $\overline{\mathcal{J}}_{g,n}^d(\sigma)$ introduced by Pagani and Tommasi, which depend nontrivially on degree $d$ and a combinatorial stability condition $\sigma$. These spaces serve as proper nonsingular Deligne–Mumford stacks that extend the classical family $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$ of universal Jacobians to the boundary of stable curves.

One of the striking facts, emerging from the results of Migliorini--Shende--Viviani, is the cohomological independence of degree and stability condition; the cohomology of $\overline{\mathcal{J}}_{g,n}^d(\sigma)$ is invariant under changes in $d$ and $\sigma$. This is counterintuitive in light of the rich boundary combinatorics and the non-uniqueness of compactified Jacobians; distinct stability conditions lead to non-isomorphic stacks with differing stratifications and orbifold structures.

Universal Jacobians and Compactification Theory

For pointed curves $(g, n)$ with $2g-2+n > 0$, universal Jacobians $\overline{\mathcal{J}}_{g,n}^d(\sigma)$0 parameterize triples $\overline{\mathcal{J}}_{g,n}^d(\sigma)$1, where $\overline{\mathcal{J}}_{g,n}^d(\sigma)$2 is a nonsingular genus $\overline{\mathcal{J}}_{g,n}^d(\sigma)$3 curve, $\overline{\mathcal{J}}_{g,n}^d(\sigma)$4 are distinct points, and $\overline{\mathcal{J}}_{g,n}^d(\sigma)$5 is a degree $\overline{\mathcal{J}}_{g,n}^d(\sigma)$6 line bundle. For $\overline{\mathcal{J}}_{g,n}^d(\sigma)$7, isomorphisms between different degree universal Jacobians can be constructed via elementary twist maps. However, the $\overline{\mathcal{J}}_{g,n}^d(\sigma)$8-equivariance is only attained when twisting by all points simultaneously, leading to refined degree congruence constraints: the $\overline{\mathcal{J}}_{g,n}^d(\sigma)$9-equivariant isomorphism holds if and only if $d$0 or $d$1 modulo $d$2.

Compactification involves extending these families over the moduli space of stable curves $d$3, retaining desired properties like flatness and properness. Numerous constructions exist; the focus here is on Pagani--Tommasi compactified Jacobians, which depend on a choice of combinatorial stability condition $d$4. Notably, the compactified Jacobian $d$5 is generally not unique for fixed $d$6, and can be non-isomorphic as stacks for different choices of $d$7.

Cohomological Invariance: Main Results

Strong Theorems

The central result is the following:

Theorem: For all $d$8, $d$9, and Pagani--Tommasi stability conditions $\sigma$0, there is an isomorphism

$\sigma$1

of $\sigma$2-Hodge structures. If $\sigma$3 are both invariant for a subgroup $\sigma$4, this isomorphism can be made $\sigma$5-equivariant.

This theorem is immediate from [MSV21], yet the paper provides a direct combinatorial proof by pairing contributions from individual boundary strata, thus offering explicit control over the invariant and independent nature of the Hodge-Deligne polynomial.

Numerically, Hodge numbers are proven to be completely independent of $\sigma$6 and $\sigma$7:

$\sigma$8

The orbifold Euler characteristic, computed in previous work [wood24], is similarly independent:

$\sigma$9

Boundary Combinatorics and Stratifications

The space $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$0 admits a boundary stratification wherein each stratum is a torus bundle over a product of moduli spaces $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$1, modulo a finite group. While individual strata vary according to $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$2, there exists a bijection between the set of triples $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$3 indexing the strata for different stability conditions, and a careful combinatorial analysis matches Hodge contributions, establishing cohomological invariance.

Further Cohomological Structures and Contradictory Claims

While additive cohomology and Hodge numbers are invariant, the cohomology ring structure is not. The cup-product and ring structure of $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$4 depends on both $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$5 and $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$6 [BaeMSY].

To address this, a graded intrinsic cohomology ring $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$7 defined via the perverse filtration is shown to be independent of degree and stability conditions, providing a canonical ring structure isomorphic to a subquotient of $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$8.

Additionally, integrals or push-forwards of tautological classes are independent of $\mathcal{J}_{g,n}^d \to \mathcal{M}_{g,n}$9 and $\overline{\mathcal{J}}_{g,n}^d(\sigma)$0, with recent explicit descriptions found in [bae2025fourier].

Derived category equivalence of compactified Jacobians with differing stability conditions is conjectured not to hold in general due to orbifold structure differences, in contrast to results in families with appropriate nodal curves.

Finally, the extension of these invariance results to moduli spaces of stable bundles of rank $\overline{\mathcal{J}}_{g,n}^d(\sigma)$1 and degree $\overline{\mathcal{J}}_{g,n}^d(\sigma)$2 remains open, with cohomological invariance postulated only for fixed $\overline{\mathcal{J}}_{g,n}^d(\sigma)$3.

Implications and Future Directions

The invariance of Hodge numbers and orbital Euler characteristics in universal compactified Jacobians, independent of degree and combinatorial stability, strongly refines our understanding of the interplay between combinatorial and geometric data in moduli theory. In particular:

• Moduli-theoretic invariance: The cohomology of Jacobian moduli spaces retains stability under combinatorial changes, facilitating calculations and theoretical analysis of periods and mixed Hodge structures.
• Algorithmic influence: The explicit stratification method provides an effective procedure for computing Hodge–Deligne polynomials, reliant on Euler characteristics of uncompactified moduli spaces, as described in Section \ref{compute}.
• Generalization and combinatorics: The direct combinatorial proof strategy opens pathways to analyze other moduli spaces with similar stratification and stability phenomena.

The negative answer for cohomology ring invariance signals the need for refined invariants, like the perverse-filtered graded cohomology ring, which capture canonical geometric structure and yield degree- and stability-independent data. Derived equivalence and orbifold cohomology distinctions point to intricate relationships between homological invariants and boundary geometry.

Speculatively, further developments may encompass symmetric function formulations, higher-rank moduli spaces, and systematic application of stratification-based algorithms to moduli spaces of bundles and sheaves. The implications for virtual Hodge numbers, birational geometry, and Grothendieck group calculations are significant, as clarified by recent results in the Appendix.

Conclusion

This paper rigorously establishes the invariance of cohomology and Hodge numbers for universal compactified Jacobians over the moduli space of stable curves, regardless of degree and combinatorial stability. Through direct combinatorial analysis and exploitation of boundary stratifications, the rich interplay between modular data and cohomological invariants is elucidated, yielding both theoretical insight and computational practicality. The non-invariance of the cohomology ring structure emphasizes the necessity of intrinsic algebraic objects, while the exploration of derived equivalences highlights the subtlety of orbifold geometry. These results inform future directions in moduli theory, algebraic geometry, and mathematical physics, providing new tools for explicit computations and deeper understanding of universal moduli spaces.

Reference: "Universal compactified Jacobians: cohomological invariance and boundary combinatorics" (2604.18377).