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Support theorem of universal compactified Jacobians

This paper establishes a full support theorem for the relative good moduli space of the universal compactified Jacobian over $\overline{\mathcal{M}}_{g,n}$ by proving that all direct summands in its BBDG decomposition have full support and explicitly describing this decomposition through two distinct proofs involving Maulik-Shen's generalization of Ngô's theorem and the variation of stability conditions.

Original authors: Yifan Wu

Published 2026-05-06

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Yifan Wu

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 trying to understand the shape of a vast, complex landscape. In mathematics, this landscape is called the moduli space of curves ( $M_{g,n}$ ). Think of this as a giant map where every single point represents a different kind of "curved surface" (like a donut with holes, or a pretzel with handles). Some of these surfaces are perfectly smooth, while others have sharp corners or pinched points (singularities).

Now, attached to every single point on this map is a hidden, intricate structure called a Jacobian. You can think of a Jacobian as a "fingerprint" or a "shadow" of that specific curved surface. It tells you about the ways you can wrap strings or sheets around the surface.

The Problem: The Foggy Boundary

For smooth surfaces, we know exactly what these Jacobians look like. They are well-behaved, smooth, and predictable. But as you move toward the "edge" of the map (where surfaces develop sharp corners), things get messy.

When the surface has sharp corners, the Jacobian becomes a chaotic, jagged mess. It's like trying to take a clear photo of a reflection in a broken mirror. Mathematicians call this the compactified Jacobian. The big question was: Even though these shapes are jagged and broken at the edges, does the "shadow" (the Jacobian) still tell us the whole story of the entire map, or does it get lost in the cracks?

The Discovery: The Full Support Theorem

The author, Yifan Wu, proves a "Full Support Theorem." Here is the simple version of what that means:

Imagine you are shining a flashlight (representing the mathematical data) from the jagged Jacobian back onto the map of curves.

The Fear: You might worry that the light only hits the smooth parts of the map, leaving the jagged, broken edges in total darkness.
The Result: Wu proves that the light hits every single part of the map, from the smoothest center to the most broken edges. No matter how jagged the surface gets, the mathematical "shadow" still contains information about the entire landscape. Nothing is lost.

How They Did It: Two Different Paths

The paper offers two ways to prove this, like taking two different hiking trails to reach the same summit.

Trail 1: The "Group Action" Hike (The First Proof)
This path uses a powerful tool developed by other mathematicians (Ngô, Maulik, and Shen).

The Metaphor: Imagine the Jacobian as a dance floor where a group of dancers (a "group scheme") moves around. Usually, if the dance floor is too bumpy, the dancers can't move smoothly, and the math breaks.
The Trick: Wu realized that by slightly "rigidifying" (stiffening) the dance floor—fixing the wobbly parts without changing the view of the landscape—the dancers could move smoothly again.
The Result: Once the floor was stiffened, the powerful "Support Theorem" could be applied. It showed that the dance (the mathematical structure) covers the entire floor, proving the light hits everywhere.

Trail 2: The "Stability" Hike (The Second Proof)
This path is a bit more like adjusting a camera lens.

The Metaphor: Sometimes, the jagged mess happens because we are looking at the surface with a "degenerate" (blurry) setting.
The Trick: Wu showed that you can slightly tweak the "stability condition" (the focus of the lens) to make the surface smooth again, just for a moment.
The Result: Once the surface is smooth, we know the light hits everywhere (because smooth surfaces are easy to understand). Then, by slowly turning the lens back to the original "blurry" setting, the proof shows that the light never actually left the edges. The information was there all along; it was just harder to see.

The Big Picture: What Does This Mean?

The paper concludes with a beautiful formula. It says that the complex, jagged structure of these Jacobians is actually built from simple, predictable building blocks derived from the smooth parts of the map.

Invariance: Whether you choose a smooth surface or a broken one, and whether you choose one type of "fingerprint" or another, the total amount of information (the cohomology) remains exactly the same.
The Takeaway: Even in the most chaotic, broken, and singular corners of this mathematical universe, the underlying structure is robust. The "shadow" never loses the shape of the object casting it.

In short: The paper proves that no matter how broken or complex the geometric shapes get at the edges of the map, their mathematical "shadows" remain complete and cover the entire territory, ensuring that no information is ever lost in the cracks.

Technical Summary: Support Theorem of Universal Compactified Jacobians

1. Problem Statement

The paper addresses the geometric and cohomological structure of the universal compactified Jacobian $\overline{\mathcal{J}}^{d,\phi}_{g,n}$ over the moduli stack of stable marked curves $\mathcal{M}_{g,n}$ . While the cohomology of these spaces is known to be invariant under the choice of degree $d$ and non-degenerate stability conditions $\phi$ (proven by Migliorini–Shende–Viviani [MSV21]), the geometry becomes significantly more complex when $\phi$ is degenerate. In the degenerate case, the presence of strictly semistable sheaves introduces severe singularities over the boundary strata of $\mathcal{M}_{g,n}$ .

The central problem is to determine the support of the direct summands appearing in the BBDG (Beilinson–Bernstein–Deligne–Gabber) decomposition of the derived pushforward of the intersection cohomology complex, $R\overline{\pi}_* IC(\overline{\mathcal{J}}^{d,\phi}_{g,n})$ , where $\overline{\pi}$ is the forgetful map to the base. Specifically, the author seeks to prove that every direct summand has full support on the base $\mathcal{M}_{g,n}$ , extending the "support theorem" known for Hitchin fibrations and non-degenerate Jacobians to this more singular, universal setting.

2. Methodology

The paper employs two distinct methodological approaches to establish the main result:

Approach A: Generalized Ngô Support Theorem

The primary proof synthesizes techniques from the theory of weak abelian fibrations and equivariant perverse sheaves.

Rigidification and Good Moduli Spaces: Direct application of Ngô's support theorem is obstructed because the universal compactified Jacobian stack is not proper over $\mathcal{M}_{g,n}$ , and the natural translation action of the relative Picard group does not descend to the coarse moduli space due to automorphisms. To resolve this, the author works with the relative good moduli space $\overline{\mathcal{J}}^{d,\phi}_{g,n}$ , which rigidifies vertical automorphisms while preserving the base $\mathcal{M}_{g,n}$ .
Base Change: To avoid technical subtleties regarding finite quotient stacks, the morphism is base-changed along a finite étale cover $B \to \mathcal{M}_{g,n}$ (a Teichmüller level structure space), resulting in a smooth variety $B$ .
Verification of Hypotheses: The author verifies that the triple $(\widetilde{\mathcal{J}}^{d,\phi}_B, P, B)$ $(J_{B}^{d, ϕ}, P, B)$ , where $P$ $P$ is the relative Picard variety, forms a weak abelian fibration in the sense of Maulik and Shen [MS23]. This involves checking:
- Properness and pure dimension of fibers.
- Affine stabilizers for the group action.
- $\delta$ -regularity: A condition relating the codimension of strata in the base to the dimension of the affine part of the stabilizer group.
Decomposition Theorem: Using the generalized support theorem (Theorem 2.2), the author proves that any support $Z$ of the decomposition must satisfy $\text{codim } Z \leq \delta_Z$ . Combined with $\delta$ -regularity, this forces $Z$ to be the entire base.
Descent: The result is descended from the cover $B$ back to $\mathcal{M}_{g,n}$ using the formalism of the equivariant derived category.

Approach B: Variation of Stability Conditions

A second, independent proof is provided for the case $n \geq 1$ (and adapted for $n=0$ via étale neighborhoods).

Wall-Crossing: The author utilizes the wall-and-chamber structure of the stability space $V^d_{g,n}$ . For a degenerate stability condition $\phi$ , a nearby non-degenerate condition $\phi'$ is chosen.
Proper Morphism: An inclusion of moduli stacks induces a proper, surjective morphism $r: \mathcal{J}^{d,\phi'}_{g,n} \to \mathcal{J}^{d,\phi}_{g,n}$ between their relative good moduli spaces.
Reduction: Since $\mathcal{J}^{d,\phi'}_{g,n}$ is smooth (for non-degenerate $\phi'$ ), its intersection complex is a shifted constant sheaf. The known support theorem for non-degenerate Jacobians [MSV21] applies to the pushforward of this constant sheaf.
Direct Summand Argument: The intersection complex $IC(\mathcal{J}^{d,\phi}_{g,n})$ is shown to be a direct summand of the pushforward of the constant sheaf from the non-degenerate side. Consequently, the support properties of the non-degenerate case transfer to the degenerate case.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The paper proves that for integers $g, n \geq 0$ with $2g-2+n > 0$ , and for the relative good moduli space $\overline{\mathcal{J}}^{d,\phi}_{g,n}$ over $\mathcal{M}_{g,n}$ :

Full Support: Every direct summand of the derived pushforward $\overline{\pi}_* IC(\overline{\mathcal{J}}^{d,\phi}_{g,n})$ has full support on $\mathcal{M}_{g,n}$ .
Explicit Decomposition: There is an isomorphism in the bounded derived category of mixed Hodge modules on $\mathcal{M}_{g,n}$ :
$\overline{\pi}_* IC(\overline{\mathcal{J}}^{d,\phi}_{g,n}) \simeq \bigoplus_{i=0}^{2g} IC\left(\mathcal{M}_{g,n}, (\wedge^i R^1\gamma_* \mathbb{Q}_{C_{g,n}})\right)[-i+g]$
where $\gamma: C_{g,n} \to \mathcal{M}_{g,n}$ is the universal curve.

Corollaries

Coarse Moduli Space (Corollary 1.2): The full support property holds for the pushforward of the intersection complex of the coarse moduli space $J$ of $\overline{\mathcal{J}}^{d,\phi}_{g,n}$ .
Topological Invariance (Corollary 1.3): The intersection cohomology groups $IH^*(\overline{\mathcal{J}}^{d,\phi}_{g,n}, \mathbb{Q})$ are isomorphic as $\mathbb{Q}$ -Hodge structures for any integers $d, d'$ and any stability conditions $\phi, \phi'$ (degenerate or non-degenerate). This extends the known invariance of cohomology groups to the degenerate setting.

4. Significance and Claims

The paper claims to provide a full support theorem for the universal compactified Jacobian, a result that was previously obstructed by the complexity of boundary strata and the lack of a proper group action on the coarse moduli space.

Methodological Synthesis: The work successfully adapts the machinery of Ngô's support theorem (originally developed for the Hitchin fibration) to the moduli of curves. It resolves the obstruction of non-properness and automorphism groups by utilizing the relative good moduli space and equivariant derived categories.
Handling Degeneracy: The paper explicitly addresses the case of degenerate stability conditions, where the geometry is singular and strictly semistable sheaves exist. It demonstrates that despite these singularities, the topological invariance of the intersection cohomology persists.
Independence of Proofs: By providing two distinct proofs—one relying on the generalized support theorem for weak abelian fibrations and the other on variation of stability conditions—the author reinforces the robustness of the result.
Motivation for Future Work: The results motivate the question of whether the "intrinsic cohomology ring" (a degeneration of the cohomology ring structure defined for non-degenerate conditions) can be extended to the setting of degenerate stability conditions.

The author maintains a modest tone regarding the scope, noting that while the cohomology groups are invariant, the cohomology ring structure generally depends on the choices of $d$ and $\phi$ , and the paper does not resolve the ring structure for the degenerate case.