Completed volumes and the DR-cycle

Technical Summary: Completed Volumes and the DR-Cycle

Problem Statement
The paper addresses the computation of Masur-Veech volumes for moduli spaces of $k$-differentials, specifically focusing on quadratic differentials ($k=2$). While formulas exist for abelian differentials and principal strata of quadratic differentials, the general case remains challenging. Two primary strategies for computing these volumes exist:

1. Ribbon Graph Counting: Decomposing surfaces into ribbon graphs and counting integer metrics, relating these counts to Witten-Kontsevich combinatorial classes. This approach, introduced by Duriev, Goujard, and Yakovlev (DGY), defines "completed volumes" to account for overcounting due to degenerate ribbon graphs.
2. Intersection Theory: Computing volumes as top intersection numbers of tautological classes on the multi-scale compactification of the moduli space.

A specific technical gap exists in the case of $n=2$ singularities for quadratic differentials. The DGY approach encounters "cumbersome" overcounting issues in this case that were previously excluded from their main recursive formulas. Furthermore, a rigorous link between the combinatorial "completed volumes" (derived from ribbon graphs) and the geometric "DR-completed volumes" (derived from the Double Ramification cycle) had not been fully established for all signatures, particularly those with two singularities.

Methodology
The authors employ a synthesis of algebraic geometry, specifically the theory of multi-scale differentials and the Double Ramification (DR) cycle, to bridge the gap between combinatorial counting and intersection theory.

1. The DR-Locus and Generalized Multi-Scale Differentials (GMS):
   The authors work within the category of logarithmic schemes, utilizing the space of generalized multi-scale differentials, $GMS_\mu$, which serves as a locus inside the rubber moduli space. They analyze the components of this space, classifying them via level graphs $\Gamma$ that impose a Global Residue Condition (GRC) at every level passage.

2. Excess Intersection Analysis:
   To compute the intersection of the DR-cycle with the top power of the tautological class $\xi^d$ (which defines the volume), the authors perform a detailed excess intersection calculation. They determine the components of the DR-cycle that contribute non-trivially to this top intersection.

   • They identify that only two specific types of level graphs contribute: special simple star graphs and sunflower graphs.
   • They compute the normal sheaf and the Chern polynomials of the excess bundles associated with these components, specifically up to terms that are "seen" by the top power $\xi^d$.

3. Recursion and Graph Correspondence:

   • For $n \ge 4$: The authors demonstrate that the terms in the DGY recursive formula for completed volumes correspond exactly to the contributions from sunflower graphs in the DR-cycle expansion.
   • For $n = 2$: This case is technically more involved due to the presence of non-separating static edges in ribbon graphs, which lead to overcounting. The authors analyze how ribbon graphs with static edges degenerate. They establish a bijection between these degenerate configurations and special simple star graphs. They perform a detailed combinatorial count (using inclusion-exclusion and specific auxiliary functions $f_2$) to show that the "overcounted" terms in the ribbon graph approach precisely match the contributions from special simple star graphs in the DR-cycle formula.

Key Contributions and Results

1. Equivalence of Completed Volumes (Theorem 1.1):
   The primary result is the proof that the DR-completed volume (defined as the intersection of the DR-cycle with the tautological class) agrees exactly with the (Ribbon graph) completed volume defined by Duriev-Goujard-Yakovlev.
   $$\text{vol}_{DR}(\mu) = \text{vol}(\mu)$$
   This equality holds for all signatures of quadratic differentials with odd entries $m_i > -2$. The proof establishes not just a numerical equality, but a term-by-term equality between the two recursive structures.

2. Structural Description of the DR-Cycle:
   The paper provides a full structural result for the components of the DR-cycle relevant to volume computations. It proves that the intersection with the top tautological class is supported only on special star graphs and sunflower graphs (Proposition 5.4). This refines previous understandings which often focused only on star-shaped graphs.

3. Resolution of the Two-Singularity Case:
   The authors complete the work of DGY by solving the technically difficult case of strata with two singularities ($n=2$). They show that the "overcounting" in the Kontsevich polynomial approach for this case arises from ribbon graphs admitting integer metrics that degenerate to zero-length edges (static edges). These configurations are shown to correspond to sunflower graphs with a non-separating static edge, which map to special simple star graphs in the DR-cycle framework.

4. Explicit Recursive Formula:
   The paper derives a unified recursive formula for the completed volume:
   $$\text{vol}_{DR}(\mu) = \text{vol}(\mu) + \sum_{\Gamma \in SF(\mu) \cup SSStar(\mu)} \frac{1}{\text{Aut}(\Gamma) k^{h_{ab}}} \prod_{e \in E(\Gamma)} \kappa_e \prod_{v \in V(\Gamma)} \text{vol}(\mu_v)$$
   where the sum runs over sunflower graphs ($SF$) and special simple star graphs ($SSStar$).

Significance and Claims
The paper claims to provide a rigorous geometric foundation for the combinatorial "completed volumes" introduced by DGY. By proving that these volumes are equal to intersection numbers on the DR-cycle, the authors:

• Validate the DGY approach as a correct method for computing Masur-Veech volumes via intersection theory.
• Offer a new pathway to investigate the large genus asymptotic conjectures (specifically [ADGZZ20]). Since the DR-completed volume is expressible via Chiodo classes (which have known asymptotic behaviors), this equivalence allows researchers to use either the ribbon graph recursion or the Chiodo class expression to derive large genus limits.
• Completes the technical landscape for quadratic differentials, specifically resolving the previously open and "cumbersome" case of two singularities.

The authors remain modest regarding the immediate solution of the large genus conjecture, noting that while the equivalence opens "a two-fold way" to approach it, the actual derivation of the limit requires further work to invert the recursion or evaluate the Chiodo coefficients in the large genus limit. They also note that the DR-completed volume is defined for any $k$ and $\mu$, exhibiting piecewise polynomial behavior that may offer further computational advantages.