Lefschetz filtration and Perverse filtration on the compactified Jacobian

This paper proves that the Lefschetz filtration and the perverse filtration on the cohomology of the compactified Jacobian of a complex integral curve with planar singularities are opposite to each other, thereby confirming a conjecture by Maulik and Yun.

The Gist

Imagine you are trying to understand the shape of a very strange, crumpled piece of paper. In the world of mathematics, this "paper" is a geometric object called a Compactified Jacobian (let's call it $J$). It's built from a curve (a line or loop) that has some sharp kinks or self-intersections (singularities).

Mathematicians love to study these shapes by looking at their "cohomology," which is essentially a way of counting and categorizing the holes, loops, and higher-dimensional features of the shape. But because our shape $J$ is crumpled and singular, it's hard to see its true structure.

This paper, by Yao Yuan, solves a puzzle about two different ways of organizing the information about this shape. The author proves that these two ways are actually perfect opposites of each other, like two sides of a coin that fit together perfectly.

Here is the breakdown using simple analogies:

1. The Two Ways of Sorting the Data

Imagine you have a giant library of books (the cohomology data) about this crumpled shape. You want to organize them into shelves.

Method A: The "Lefschetz" Filter (The Gravity Filter)

• The Concept: Imagine you have a heavy weight (an "ample divisor," let's call it $\Theta$) that you drop onto the library.
• How it works: When you drop this weight, it pushes the books down. Some books are heavy and sink to the bottom shelves; others are light and stay on the top.
• The Result: This creates a filtration (a sorting system) where books are grouped by how "deep" they sink. This is called the Lefschetz filtration. It's based on the physical weight of the geometry.

Method B: The "Perverse" Filter (The Family Tree Filter)

• The Concept: Now, imagine you don't just look at the crumpled shape in isolation. Instead, you imagine this shape is part of a larger family of shapes that change smoothly over time (a "family of curves").
• How it works: You look at how the shape behaves as it morphs through this family. You ask: "At what stage of the family's history did this specific feature appear?"
• The Result: This creates a different sorting system called the Perverse filtration. It organizes the books based on their "history" or "complexity" within the family, rather than their weight.

2. The Big Question

For a long time, mathematicians (specifically Maulik and Yun) suspected that these two sorting methods were opposites.

• If a book is on the very bottom shelf of the "Gravity Filter" (Lefschetz), it should be on the very top shelf of the "History Filter" (Perverse).
• If a book is in the middle of one, it should be in the middle of the other, but in reverse order.

Think of it like a stack of pancakes.

• Filter A sorts them by size (biggest at the bottom).
• Filter B sorts them by age (oldest at the top).
• The conjecture was: The biggest pancakes are also the oldest, and the smallest are the youngest.

3. The Author's Solution

Yao Yuan proves that this "Opposite Conjecture" is true.

How did he do it? (The Magic Trick)
To prove this, the author used a mathematical tool called the Fourier Transform.

• The Analogy: Imagine you have a song. The "Gravity Filter" listens to the song's volume (loud vs. quiet). The "History Filter" listens to the song's pitch (high vs. low notes).
• Usually, volume and pitch are unrelated. But in this specific mathematical world, the Fourier Transform acts like a magic translator that converts "Volume" into "Pitch."
• Yuan showed that if you take the "Gravity" sorting, translate it using this magic tool, and then translate it back, you get the exact "History" sorting, but flipped upside down.

4. Why Does This Matter?

You might ask, "Who cares about crumpled paper and pancake sorting?"

• It's a Bridge: This result connects two very different areas of math: one that looks at shapes physically (Lefschetz) and one that looks at how shapes evolve in families (Perverse).
• It Solves a Mystery: It confirms that the universe of these geometric shapes has a hidden, perfect symmetry. The "weight" of a shape and its "complexity" are two sides of the same coin.
• It Helps with Singularities: Most real-world shapes aren't perfect smooth spheres; they have kinks and tears. This paper gives us a new, powerful way to understand the cohomology (the "DNA") of these messy, singular shapes.

Summary in One Sentence

Yao Yuan proved that for a specific type of crumpled geometric shape, the way you sort its features by "weight" is exactly the reverse of how you sort them by "complexity," and he used a mathematical magic trick (the Fourier Transform) to show that these two views are perfectly linked.

Technical Summary

Here is a detailed technical summary of the paper "Lefschetz Filtration and Perverse Filtration on the Compactified Jacobian" by Yao Yuan.

1. Problem Statement and Context

The paper addresses a conjecture by Maulik and Yun regarding the relationship between two distinct filtrations on the cohomology group $H^*(J)$ of the compactified Jacobian $J$ of a complex integral curve $C$ with planar singularities.

• The Object: Let $C$ be a singular curve. The compactified Jacobian $J$ parametrizes rank-1 torsion-free sheaves on $C$. Unlike the Jacobian of a smooth curve, $J$ is generally singular.
• Filtration 1: The Lefschetz Filtration ($W$): Defined via the nilpotent endomorphism $L_\Theta$ induced by the cup product with a specific ample divisor class $\Theta$ (a generalized theta divisor). This filtration arises from the Hard Lefschetz theorem context, though $J$ is singular, so standard Hard Lefschetz properties do not automatically hold.
• Filtration 2: The Perverse Filtration ($P$): Defined by embedding $C$ into a smooth family of curves $\mathcal{C} \to \mathcal{B}$ and considering the relative compactified Jacobian $\mathcal{J} \to \mathcal{B}$. By the Decomposition Theorem for perverse sheaves, the derived pushforward $Rf_* \mathbb{Q}_{\mathcal{J}}$ decomposes into shifted perverse cohomologies. Restricting this to the fiber $J$ yields a filtration $P_{\leq i}$.
• The Conjecture: Maulik and Yun conjectured that these two filtrations are opposite to each other. Specifically, if $W_k$ is the Lefschetz filtration and $P_{\leq m}$ is the perverse filtration, their intersection should vanish when $m + k < 2g$ (where $g$ is the arithmetic genus).

2. Methodology

The author employs a combination of geometric representation theory, bivariant homology theory, and Fourier-Mukai transforms adapted to singular varieties.

A. Rennemo's Decomposition

The paper builds upon the work of Rennemo, who established a decomposition of $H^*(J)$ based on the Hilbert scheme of points $C^{[n]}$ on the curve.

• Rennemo defined operators $\mu_{\pm}[pt]$ and $\mu_{\pm}[C]$ acting on the homology of Hilbert schemes.
• This induces a grading (the D-decomposition) on $H^*(J)$: $H^*(J) = \bigoplus_{k \geq 0} D_k H^*(J)$.
• It was previously known that the perverse filtration corresponds to the partial sums of this grading: $P_{\leq m} H^*(J) = \bigoplus_{k \leq m} D_k H^*(J)$.

B. The Challenge of Singularities

Standard tools for smooth varieties (like the classical Fourier transform defined via a line bundle on $J \times J$) fail because $J$ is singular and the Poincaré sheaf $\mathcal{P}$ on $J \times J$ is only a maximal Cohen-Macaulay (MCM) sheaf, not a line bundle. Consequently, the Chern character is not well-defined in the usual sense, and intersection products are problematic.

C. Bivariant Theory and Generalized Fourier Transform

To overcome the singularity issues, Yuan utilizes Fulton's bivariant homology theory:

1. Bivariant Classes: Instead of standard cohomology, the author works with bivariant groups $H^*(X \xrightarrow{f} Y)$. This allows for well-defined pushforwards and pullbacks even when maps are not smooth or proper in the classical sense.
2. Generalized Chern Character: Using the Riemann-Roch for singular varieties (specifically the $\tau$-class), the author defines a "Chern character" $\text{ch}_\tau(\mathcal{P})$ for the MCM sheaf $\mathcal{P}$.
3. Fourier Transform ($\mathcal{F}$): A Fourier transform $\mathcal{F}: H^*(J) \to H^*(J)$ is constructed using the bivariant class of $\mathcal{P}$:
   $$\mathcal{F}(\alpha) = q_{1,*}(\text{eq}_2^*(\alpha) \cdot \text{ch}_\tau(\mathcal{P}))$$
   The author proves that $\mathcal{F}$ is an isomorphism and admits an inverse $\mathcal{F}_1$ constructed via the dual sheaf $\mathcal{P}^\vee$.

D. Construction of the $\mathfrak{sl}_2$-Triple

The core of the proof involves constructing an $\mathfrak{sl}_2$-triple $(e, h, f)$ acting on $H^*(J)$:

• $e$: Defined as the operator of cupping with the theta divisor $\Theta$ (acting on cohomology via the dual map $e^\vee$).
• $f$: Defined via the Fourier transform: $f = -\mathcal{F} \circ e \circ \mathcal{F}^{-1}$.
• $h$: The commutator $[e, f]$.

The author demonstrates that these operators satisfy the $\mathfrak{sl}_2$ relations and that the eigenspace decomposition of the operator $h$ coincides exactly with Rennemo's $D$-decomposition.

3. Key Contributions and Results

Main Theorem (Theorem 1.3 / Corollary 3.7)

The paper proves that the operators $(e^\vee, [e^\vee, f^\vee], f^\vee)$ form an $\mathfrak{sl}_2$-triple on $H^*(J)$. Crucially, the eigenspace decomposition of the Cartan element corresponds to the $D$-decomposition:
$$H^*(J) = \bigoplus_{k} D_k H^*(J)$$
where $D_k H^*(J)$ is the eigenspace with eigenvalue $k - g$.

Consequence: Opposite Filtrations

The existence of this $\mathfrak{sl}_2$-triple implies a specific relationship between the filtrations:

1. Lefschetz Filtration: $W_k H^*(J) = \bigoplus_{j \geq 2g-k} D_j H^*(J)$.
2. Perverse Filtration: $P_{\leq m} H^*(J) = \bigoplus_{j \leq m} D_j H^*(J)$.
3. Opposition: The intersection $W_k \cap P_{\leq m} = 0$ whenever $m + k < 2g$.

This confirms Conjecture 2.17 of Maulik and Yun for complex curves with planar singularities.

Technical Lemmas

• Proposition 2.4: Establishes that the Chern classes of a specific vector bundle $E_n$ on $J$ are powers of the theta class $\Theta$ (specifically $c_i(E_n) = \frac{1}{i!}(-\Theta)^i$). This is proven using a resolution of singularities and properties of the universal sheaf.
• Proposition 3.2 & 3.3: Rigorously define the Fourier transform in the bivariant setting and prove it is an isomorphism with a computable inverse, overcoming the lack of smoothness on $J$.

4. Significance

• Resolution of a Major Conjecture: The paper provides a definitive proof of the duality between the geometric Lefschetz filtration and the topological perverse filtration for compactified Jacobians of singular curves.
• Extension of Fourier-Mukai Theory: It successfully generalizes the classical Fourier-Mukai transform (which usually requires smoothness or abelian varieties) to the setting of singular compactified Jacobians using bivariant homology. This opens the door for applying similar techniques to other moduli spaces of sheaves on singular varieties.
• Representation Theoretic Insight: By explicitly constructing the $\mathfrak{sl}_2$-triple, the paper reveals a deep representation-theoretic structure underlying the cohomology of singular moduli spaces, linking the geometry of the theta divisor to the perverse sheaf decomposition.
• Methodological Framework: The use of bivariant theory to handle intersection products and Chern characters on singular spaces serves as a robust template for future research in algebraic geometry involving singular moduli problems.

In summary, Yao Yuan's work bridges the gap between the geometry of singular curves and the topology of their Jacobians, proving that the "hard" geometric filtration (Lefschetz) and the "soft" topological filtration (Perverse) are perfectly dual, mediated by a generalized Fourier transform.