Dualizable abelian fibrations

1. Problem Statement and Motivation

The Core Problem:
The paper addresses the structural understanding of the Decomposition Theorem for abelian fibrations $f: M \to B$ (proper surjective morphisms where general fibers are abelian varieties). While the Decomposition Theorem (Beilinson-Bernstein-Deligne-Gabber) guarantees that the derived pushforward $Rf_* \mathbb{Q}_M$ splits into a direct sum of shifted simple perverse sheaves, determining the specific supports and local systems is generally difficult.

The Gap:

• Abelian Schemes: For smooth abelian schemes (families of abelian varieties), Beauville and Deninger-Murre established a rich structure: a motivic decomposition (Beauville decomposition), a Fourier-Mukai duality, and a multiplicative structure on cohomology compatible with the decomposition.
• Singular Fibers: When fibers are singular (e.g., compactified Jacobians of nodal curves or Hitchin systems), the "multiplication by $N$" maps used in the smooth case often fail to extend, causing the Beauville decomposition and multiplicative structures to break down.
• Ngô's Contribution: Ngˆo introduced $\delta$-regular weak abelian fibrations, utilizing a group scheme action to determine supports. However, this did not fully recover the multiplicative and motivic structures seen in the smooth case.

The Goal:
The authors aim to define a class of fibrations called Dualizable Abelian Fibrations. This class generalizes the duality structure of abelian schemes to the singular setting, allowing for the recovery of motivic decompositions, multiplicative perverse filtrations, and isomorphisms between cohomology rings of different fibrations sharing a dual.

2. Methodology and Definitions

The authors construct a framework based on the existence of a dual fibration and a Poincaré sheaf that behaves like the Poincaré line bundle in the smooth case, even over singular loci.

2.1. Definition of Dualizable Abelian Fibration

An abelian fibration $f: M \to B$ of relative dimension $g$ is dualizable if it satisfies four axioms:

• Axiom A (Dual Existence): There exists a dual fibration $f^\vee: M^\vee \to B$ such that over a Zariski open set $U \subset B$, $M_U$ and $M^\vee_U$ are dual abelian schemes (i.e., $M^\vee_U \cong \text{Pic}^0(M_U)$).
• Axiom B (Full Support): The decomposition theorem for both $Rf_* \mathbb{Q}_M$ and $Rf^\vee_* \mathbb{Q}_{M^\vee}$ has full support (all simple perverse sheaves have support equal to the entire base $B$). This is closely related to Ngˆo's $\delta$-regularity condition.
• Axiom C (Extension of Poincaré Bundle): The normalized Poincaré line bundle $P_U$ on $M^\vee_U \times_U M_U$ extends to an object $P \in D^b\text{Coh}(M^\vee \times_B M)$ such that $P$ and its inverse $P^{-1}$ induce Fourier-Mukai equivalences (or at least satisfy $P^{-1} \circ P \simeq \mathcal{O}_{\Delta}$).
• Axiom C+ (Support Condition): The objects $P^{-1} \circ P^{\otimes N}$ and $P^{\otimes N} \circ P^{-1}$ have support of codimension $g$ for infinitely many $N$. This serves as a substitute for the "multiplication by $N$" map in the singular setting.
• Axiom D (Convolution Extension): A specific object $K$ constructed from $P$ and the small diagonal is supported in codimension $g$, and its support contains the closure of the graph of the multiplication map on the smooth locus as its unique pure codimension $g$ component. This allows for a convolution product on the dual side.

2.2. Key Examples

The paper identifies several geometric objects satisfying these axioms:

1. Compactified Jacobians: For integral locally planar curves, the relative compactified Jacobian $J_C \to B$ is self-dual ($f = f^\vee$).
2. Universal Fine Compactified Jacobians: Over the moduli space of stable curves $\overline{M}_{g,n}$, these are dualizable, with the dual depending on the choice of stability condition.
3. Parabolic Hitchin Systems: Unlike the standard Hitchin system (which fails Axiom B), the parabolic Hitchin system associated with a general residue is dualizable.

3. Key Contributions and Results

The theory of dualizable abelian fibrations yields three major structural results that generalize the smooth case:

3.1. Motivic Decomposition (Theorem 3.2a)

• Result: For any dualizable abelian fibration, the Corti-Hanamura Motivic Decomposition Conjecture holds.
• Mechanism: The Fourier transforms induced by $P$ and $P^{-1}$ define orthogonal projectors $q_i$ on the relative diagonal. These projectors decompose the relative Chow motive $h(M)$ into a direct sum $\bigoplus h_i(M)$.
• Significance: This provides a canonical motivic lifting of the Decomposition Theorem, even in the presence of singular fibers.

3.2. Multiplicativity of Perverse Filtrations (Theorem 3.2b, 3.3a)

• Result: The perverse filtration $P_\bullet H^*(M, \mathbb{Q})$ is multiplicative with respect to the cup product:
  $$\cup: P_i H^*(M) \times P_j H^*(M) \to P_{i+j} H^*(M)$$
• Significance: This resolves a long-standing question. While the Decomposition Theorem is not always multiplicative for singular fibrations (e.g., nodal curves), the perverse filtration (which is the sheaf-theoretic shadow of the motivic decomposition) retains the multiplicative structure. This generalizes results by Oblomkov-Yun for specific singular curves to all integral locally planar curves.

3.3. Isomorphism of Cohomology Rings via Duality (Theorem 3.2c, 3.3b)

• Result: If two fibrations $f_1: M_1 \to B$ and $f_2: M_2 \to B$ share a common dual $f^\vee: M^\vee \to B$, their associated graded cohomology rings (with respect to the perverse filtration) are isomorphic as bigraded algebras:
  $$\text{Gr}^P H^*(M_1) \cong \text{Gr}^P H^*(M_2)$$
• Mechanism: Both rings are isomorphic to the convolution algebra on the cohomology of the dual $M^\vee$.
• Significance: This explains why different compactifications (e.g., different stability conditions for Jacobians) yield isomorphic "intrinsic" cohomology rings, even if their raw cohomology rings differ.

3.4. Applications to Major Conjectures

The framework provides proofs for several deep conjectures:

• P = W Conjecture: Proven for the Hitchin system by reducing it to the parabolic Hitchin system (which is dualizable) and using the Fourier transform to locate tautological classes in the perverse filtration.
• P = C Conjecture: Proven for moduli spaces of sheaves on del Pezzo surfaces and free ranges of Hitchin systems. It asserts that the position of a tautological class in the perverse filtration is determined by its Chern grading.
• Chern Classes: The authors prove that for dualizable fibrations, Chern classes of the tangent bundle satisfy $c_i(TM) \in P_i H^{2i}(M)$, generalizing results for abelian schemes.

4. Significance and Impact

1. Unification of Singular and Smooth Geometry: The paper successfully extends the powerful tools of Fourier-Mukai duality (previously restricted to smooth abelian schemes) to a broad class of singular fibrations. It identifies the precise geometric conditions (Axioms A-D) required to maintain this structure.
2. Resolution of Open Problems: It provides a unified framework to prove the P=W and P=C conjectures, which were previously approached via disparate methods (representation theory, Higgs bundles, etc.).
3. Intrinsic Structures: It reveals that while the cohomology ring of a singular fibration may depend on auxiliary choices (like compactification or stability conditions), the associated graded ring with respect to the perverse filtration is an intrinsic invariant. This suggests a deeper "motivic" stability in these moduli spaces.
4. New Tools for Langlands Program: By clarifying the decomposition theorem for Hitchin-like systems, the work reinforces the geometric foundations of the Langlands program, specifically the fundamental lemma and its generalizations.

In summary, Maulik, Shen, and Yin establish that duality is the key structural feature that restores the rich algebraic and motivic properties of abelian fibrations in the presence of singularities, offering a robust framework for future research in algebraic geometry and mathematical physics.