[Paper Review] Birational invariance in logarithmic Gromov-Witten theory

Dan Abramovich, Jonathan Wise|arXiv (Cornell University)|Jun 5, 2013

Geometric and Algebraic Topology3 citations

TL;DR

This paper establishes birational invariance for logarithmic Gromov–Witten invariants under logarithmic modifications between logarithmically smooth varieties. Using obstruction theory and the geometry of Artin fans, the authors prove that the virtual fundamental class of the moduli space of logarithmic stable maps is preserved under such modifications, implying that invariants with primary insertions from the base scheme are identical on the source and target. This confirms a conjecture of Mark Gross and extends birational invariance to the logarithmic setting.

ABSTRACT

Gromov-Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. In this note we show that logarithmic Gromov-Witten invariants are also invariant under appropriately defined logarithmic modifications.

Motivation & Objective

• To resolve a conjecture by Mark Gross on the behavior of logarithmic Gromov–Witten invariants under logarithmic modifications.
• To establish that logarithmic Gromov–Witten invariants are invariant under proper, birational, logarithmically étale morphisms between logarithmically smooth varieties.
• To demonstrate that the virtual fundamental class of the moduli space of logarithmic stable maps is preserved under such modifications.
• To extend the notion of birational invariance from classical Gromov–Witten theory to the logarithmic setting using obstruction theory and algebraic stacks.

Proposed method

• Constructs the moduli stack M(X) of logarithmic stable maps to a logarithmic scheme X, equipped with a virtual fundamental class [M(X)]vir.
• Defines logarithmic modifications as proper, birational, logarithmically étale morphisms h: Y → X, and studies the induced morphism π: M(Y) → M(X).
• Uses the theory of Artin fans and logarithmic algebraic stacks to analyze the geometry of M′(Y → X), the fiber product of M(Y) and M(X) over the log stack Log(M).
• Applies obstruction theory techniques to show that the relative obstruction theories for M(X) over Log(M) and M(Y) over M′(Y → X) are compatible via pullback.
• Establishes that the perfect obstruction theory for M(X) pulls back to that of M(Y), ensuring compatibility of virtual fundamental classes.
• Applies Costello’s theorem on virtual pullbacks to conclude that h∗[M(Y)]vir = [M(X)]vir.

Experimental results

Research questions

• RQ1 Does the virtual fundamental class of the moduli space of logarithmic stable maps remain invariant under logarithmic modifications?
• RQ2 How do logarithmic Gromov–Witten invariants transform under proper, birational, logarithmically étale morphisms between logarithmically smooth varieties?
• RQ3 Can birational invariance in Gromov–Witten theory be extended to the logarithmic setting?
• RQ4 What is the relationship between the obstruction theories of M(X) and M(Y) under a logarithmic modification h: Y → X?

Key findings

• The virtual fundamental class of the moduli space M(Y) of logarithmic stable maps pulls back to the virtual fundamental class of M(X), i.e., h∗[M(Y)]vir = [M(X)]vir.
• The induced morphism π: M(Y) → M(X) is virtually birational, meaning it induces an isomorphism on dense open substacks with trivial logarithmic structures.
• The obstruction theory for M(X) over Log(M) pulls back to the obstruction theory for M(Y) over M′(Y → X), ensuring compatibility of virtual cycles.
• For any numerical data ΓX on X, there exists a unique ΓY on Y such that the invariants ⟨α1⋯αn⟩XΓX and ⟨h∗α1⋯h∗αn⟩YΓY coincide, while all other lifts vanish.
• The result holds only for toroidal (logarithmic) modifications, not for arbitrary birational morphisms, such as non-toric blowups of P².
• The key technical step is identifying the obstruction theories via pullback along the contraction of rational curves in the base change C = C ×X Y.