[Paper Review] A tale of two moduli spaces: logarithmic and multi-scale differentials

Dawei Chen, Samuel Grushevsky+3 more|arXiv (Cornell University)|Dec 9, 2022

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes an isomorphism between the coarse moduli stacks of logarithmic rubber maps and generalized multi-scale differentials for stable curves with prescribed zero and pole orders, proving their equivalence modulo the global residue condition. It further shows that both spaces arise as blowups of the incidence variety compactification (or normalization of the Deligne–Mumford compactification in higher genus), thereby establishing their projectivity and providing a refined double ramification cycle formula in the twisted Hodge bundle.

ABSTRACT

Multi-scale differentials were constructed by M.~Bainbridge, D.~Chen, Q.~Gendron, S.~Grushevsky, and M.~Möller, from the viewpoint of flat and complex geometry, for the purpose of compactifying moduli spaces of curves together with a differential with prescribed orders of zeros and poles. Logarithmic differentials were constructed by S.~Marcus and J.~Wise, as a generalization of stable rubber maps from Gromov--Witten theory. Modulo the global residue condition that isolates the main components of the compactification, we show that these two kinds of differentials are equivalent, and establish an isomorphism of their (coarse) moduli stacks. Moreover, we describe the rubber and multi-scale spaces as an explicit blowup of the moduli space of stable pointed rational curves in the case of genus zero, and as a global blowup of the incidence variety compactification for arbitrary genera, which implies their projectivity. We also propose a refined double ramification cycle formula in the twisted Hodge bundle which interacts with the universal line bundle class.

Motivation & Objective

• To resolve an open problem in [BCGGM19] by defining a smooth moduli stack of generalized multi-scale differentials without relying on Teichmüller markings.
• To unify two distinct compactification frameworks—logarithmic rubber maps and multi-scale differentials—by proving their equivalence under the global residue condition.
• To describe the coarse moduli spaces of both rubber and multi-scale differentials as explicit blowups of known compactifications, thereby proving their projectivity.
• To extend the double ramification cycle formula to the twisted Hodge bundle, incorporating interactions with the universal line bundle class.

Proposed method

• Constructs the moduli stack $ G\Xi\overline{\mathcal{M}}_{g,n}(\mu) $ as the coarse moduli space of generalized multi-scale differentials, omitting the global residue condition.
• Defines the logarithmic rubber space $ \mathbf{Rub}_{\mathcal{L}} $ for a line bundle $ \mathcal{L} $, with fiber over a curve $ X $ parameterizing piecewise linear functions $ \beta $ and isomorphisms $ \mathcal{O}_X(\beta) \cong \mathcal{L} $.
• Establishes an isomorphism between the main components of the logarithmic and multi-scale moduli stacks after imposing the global residue condition.
• Uses tropical geometry and graph-theoretic data (vertex functions, edge slopes, and local maximum vertices) to define local ideal sheaves $ J(\Gamma) $ that glue to a global ideal sheaf $ J $ on the incidence variety compactification.
• Shows that the main component of the multi-scale space is the normalization of the blowup of the incidence variety compactification along $ J $, and proves projectivity via this construction.
• Applies universal properties of blowups and Hodge bundles to construct inverse maps and verify the blowup description globally.

Experimental results

Research questions

• RQ1 Are the logarithmic rubber maps and generalized multi-scale differentials equivalent as moduli spaces when the global residue condition is imposed?
• RQ2 Can the coarse moduli spaces of both rubber and multi-scale differentials be described as explicit blowups of known compactifications of $ \overline{\mathcal{M}}_{g,n} $?
• RQ3 What is the precise relationship between the incidence variety compactification and the moduli space of multi-scale differentials?
• RQ4 How does the double ramification cycle refine in the twisted Hodge bundle when interacting with the universal line bundle class?
• RQ5 Can the blowup description be extended from genus zero to arbitrary genera using the normalization of the Deligne–Mumford compactification?

Key findings

• The coarse moduli space of the main component of multi-scale differentials is isomorphic to the normalization of the blowup of the incidence variety compactification along a globally defined ideal sheaf $ J $, proving its projectivity.
• For genus zero, the rubber and multi-scale spaces are explicitly described as blowups of the moduli space of stable pointed rational curves.
• In arbitrary genus, the rubber and multi-scale spaces are shown to be global blowups of the normalization of the closure of the stratum in the Deligne–Mumford compactification.
• The main component of the multi-scale moduli space is the normalization of the blowup of the incidence variety compactification along the ideal sheaf $ J $, which is constructed from local data on dual graphs.
• The paper proves that the logarithmic rubber space and the generalized multi-scale differential space are isomorphic after imposing the global residue condition, resolving a key equivalence question.
• A refined double ramification cycle formula is proposed in the twisted Hodge bundle, which interacts with the universal line bundle class via the blowup structure.