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Moduli spaces of twisted maps to smooth pairs

Robert Crumplin

TL;DR

The paper develops a universal moduli space ${\mathrm{Orb}}_{\Lambda}(\mathcal{A}_r)$ for twisted maps to smooth pairs, encoding tangency data via line bundle–section data on twisted curves. It provides a tropical/combinatorial classification of irreducible components in terms of inducible essential mod $r$ tropical types, and constructs free semi-abelian actions that organize these components into torsors over boundary strata. It proves that degrees of comparison maps between rooting parameters are monomials in the extra parameter with exponent bounded by $\max(0,2g-1)$, and formulates a precise polynomiality framework through $\widehat{\mathbb{Z}}$-tropical types, canonical liftings, and stratified Chow theory. In genus 1 the virtual class matches the usual fundamental class on finite-type open substacks, recovering Tseng–You-type polynomiality phenomena and enabling a path to relate universal to geometric invariants for smooth pairs; the paper concludes with concrete geometric examples, including Maulik’s setup, illustrating the machinery and its implications for higher-genus comparisons and potential split-virtuallity structures.

Abstract

We study moduli spaces of twisted maps to a smooth pair in arbitrary genus, and give geometric explanations for previously known comparisons between orbifold and logarithmic Gromov--Witten invariants. Namely, we study the space of twisted maps to the universal target and classify its irreducible components in terms of combinatorial/tropical information. We also introduce natural morphisms between these moduli spaces for different rooting parameters and compute their degree on various strata. Combining this with additional hypotheses on the discrete data, we show these degrees are monomial of degree between $0$ and $\max(0,2g-1)$ in the rooting parameter. We discuss the virtual theory of the moduli spaces, and relate our polynomiality results to work of Tseng and You on the higher genus orbifold Gromov--Witten invariants of smooth pairs, recovering their results in genus $1$. We discuss what is needed to deduce arbitrary genus comparison results using the previous sections. We conclude with some geometric examples, starting by re-framing the original genus $1$ example of Maulik in this new formalism.

Moduli spaces of twisted maps to smooth pairs

TL;DR

The paper develops a universal moduli space for twisted maps to smooth pairs, encoding tangency data via line bundle–section data on twisted curves. It provides a tropical/combinatorial classification of irreducible components in terms of inducible essential mod tropical types, and constructs free semi-abelian actions that organize these components into torsors over boundary strata. It proves that degrees of comparison maps between rooting parameters are monomials in the extra parameter with exponent bounded by , and formulates a precise polynomiality framework through -tropical types, canonical liftings, and stratified Chow theory. In genus 1 the virtual class matches the usual fundamental class on finite-type open substacks, recovering Tseng–You-type polynomiality phenomena and enabling a path to relate universal to geometric invariants for smooth pairs; the paper concludes with concrete geometric examples, including Maulik’s setup, illustrating the machinery and its implications for higher-genus comparisons and potential split-virtuallity structures.

Abstract

We study moduli spaces of twisted maps to a smooth pair in arbitrary genus, and give geometric explanations for previously known comparisons between orbifold and logarithmic Gromov--Witten invariants. Namely, we study the space of twisted maps to the universal target and classify its irreducible components in terms of combinatorial/tropical information. We also introduce natural morphisms between these moduli spaces for different rooting parameters and compute their degree on various strata. Combining this with additional hypotheses on the discrete data, we show these degrees are monomial of degree between and in the rooting parameter. We discuss the virtual theory of the moduli spaces, and relate our polynomiality results to work of Tseng and You on the higher genus orbifold Gromov--Witten invariants of smooth pairs, recovering their results in genus . We discuss what is needed to deduce arbitrary genus comparison results using the previous sections. We conclude with some geometric examples, starting by re-framing the original genus example of Maulik in this new formalism.
Paper Structure (40 sections, 43 theorems, 217 equations, 3 figures)

This paper contains 40 sections, 43 theorems, 217 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. Main results
  4. History of comparison results and future directions.
  5. Acknowledgements
  6. Irreducible components of the universal space of twisted maps
  7. Set up
  8. Contact data and the moduli space
  9. Tropical types
  10. Distinguished loci of $\mathrm{Orb}_{\Lambda}(\mathcal{A}_r)$
  11. Structure of $\mathrm{Orb}_{\Lambda}(\mathcal{A}_r)$
  12. Set up
  13. Structure of the line bundles
  14. Semi-abelian group actions
  15. Classification of irreducible components
  16. ...and 25 more sections

Key Result

Theorem 1

There is a bijection given by

Figures (3)

  • Figure 1: Two irreducible components of $\mathrm{Orb}_{\Lambda}(\mathcal{A}_r)$ in genus $1$.
  • Figure 2: A genus $1$ pre-tropical type.
  • Figure 3: Three different depictions of mod $r$ tropical types

Theorems & Definitions (115)

  • Theorem 1: Theorem \ref{['irredcpts']}
  • Proposition 2: = Proposition \ref{['action']}
  • Theorem 3: = Theorem \ref{['degrees']} + Theorem \ref{['essential degree theorem']} + Proposition \ref{['aut restriction image']}
  • Theorem 4: = Theorem \ref{['essential degree theorem']} + Corollary \ref{['essential polynomial']}
  • Theorem 5: = Corollary \ref{['g = 1 virtual class']} + Theorem \ref{['g = 1 virtual polynomial']}
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 105 more