[Paper Review] Gromov-Witten theory via roots and logarithms

Luca Battistella, Navid Nabijou+1 more|arXiv (Cornell University)|Mar 31, 2022

Polynomial and algebraic computation4 citations

TL;DR

This paper establishes a convergence of orbifold Gromov–Witten theory via multi-root stacks to logarithmic Gromov–Witten theory for simple normal crossings pairs $(X|D)$ in genus zero. By introducing a $ackslash$Lambda$-sensitive blowup condition and using a naive Gromov–Witten theory as an intermediary, the authors prove that sufficiently refined strata blowups yield orbifold invariants isomorphic to the logarithmic invariants, thereby identifying birational invariance as the key mechanism distinguishing the two theories.

ABSTRACT

Orbifold and logarithmic structures provide independent routes to the virtual enumeration of curves with tangency orders for a simple normal crossings pair $(X|D)$. The theories do not coincide and their relationship has remained mysterious. We prove that the genus zero orbifold theories of multi-root stacks of strata blowups of $(X|D)$ converge to the corresponding logarithmic theory of $(X|D)$. With fixed numerical data, there is an explicit combinatorial criterion that guarantees when a blowup is sufficiently refined for the theories to coincide. There are two key ideas in the proof. The first is the construction of a naive Gromov-Witten theory, which serves as an intermediary between roots and logarithms. The second is a smoothing theorem for tropical stable maps; the geometric theorem then follows via virtual intersection theory relative to the universal target. The results import new computational tools into logarithmic Gromov-Witten theory. As an application, we show that the genus zero logarithmic Gromov-Witten theory of a pair is determined by the absolute Gromov-Witten theories of its strata.

Motivation & Objective

• To resolve the long-standing mystery of the relationship between logarithmic and orbifold Gromov–Witten theories for simple normal crossings pairs $(X|D)$.
• To identify the precise geometric condition under which orbifold invariants of multi-root stacks converge to logarithmic invariants.
• To establish a reconstruction principle for genus zero logarithmic Gromov–Witten invariants using absolute invariants of the strata of $D$.
• To provide a computational bridge from logarithmic to orbifold invariants via a new intermediate theory—naive Gromov–Witten theory.
• To demonstrate that birational invariance is the essential property distinguishing logarithmic from orbifold theories.

Proposed method

• Introduce a new 'naive' Gromov–Witten theory as a bridge between orbifold and logarithmic theories.
• Define a combinatorial condition—$ackslash$Lambda$-sensitivity—on iterated blowups of $(X|D)$ along strata to ensure convergence of orbifold and logarithmic invariants.
• Construct a smoothing theorem for tropical stable maps to relate the virtual cycles of the naive and logarithmic theories.
• Use virtual intersection theory relative to the universal target to lift tropical results to the geometric setting.
• Apply the quantum Lefschetz theorem for orbifolds to reduce the restricted Gromov–Witten theory of multi-root stacks to invariants of strata.
• Localize computations using torus actions on projective bundles and compute contributions via fixed-point localization and virtual pushforward theorems.

Experimental results

Research questions

• RQ1 Under what conditions do the genus zero orbifold Gromov–Witten invariants of multi-root stacks of strata blowups of $(X|D)$ coincide with the logarithmic invariants of $(X|D)$?
• RQ2 What geometric property explains the divergence between logarithmic and orbifold Gromov–Witten theories in genus zero?
• RQ3 Can the genus zero logarithmic Gromov–Witten theory of $(X|D)$ be reconstructed from the absolute Gromov–Witten theories of its strata?
• RQ4 How does the naive Gromov–Witten theory serve as an intermediary between the orbifold and logarithmic approaches?
• RQ5 What role does birational invariance play in unifying the two theories?

Key findings

• The genus zero logarithmic Gromov–Witten invariants of $(X|D)$ with numerical data $ackslash$Lambda$ are isomorphic to the orbifold invariants of any $ackslash$Lambda$-sensitive blowup $(X^{ackslash$dagger$}|D^{ackslash$dagger$})\to(X|D)$.
• The multi-root orbifold theories of strata blowups stabilize to the logarithmic theory under the $ackslash$Lambda$-sensitivity condition, a non-trivial consequence of the convergence result.
• The genus zero logarithmic Gromov–Witten theory of $(X|D)$ is uniquely reconstructible from the absolute Gromov–Witten theories of all strata of $D$, generalizing a result of Maulik and Pandharipande.
• The restricted genus zero Gromov–Witten theory of a multi-root stack $ackslash$mathcal{X}$ is algorithmically determined by the absolute theories of the strata of $(X|D)$ via localization and virtual pushforward.
• The Euler class term in the localization computation matches the virtual fundamental class of the strata intersection $\cap_{i\in I}D_i$, confirming the reduction to absolute invariants.
• The virtual pushforward theorem for root gerbes enables the translation of orbifold integrals over $ackslash$mathsf{Orb}$ackslash$Lambda$}(ackslash$mathcal{Z}$ackslash$mathrm{I}$)$ to integrals over the coarse moduli space $ackslash$mathsf{M}$ackslash$Lambda$}(X)$, with insertions pulled back from $X$.