[Paper Review] Relative and orbifold Gromov-Witten invariants

Dan Abramovich, Charles Cadman+1 more|arXiv (Cornell University)|Apr 6, 2010

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes a precise correspondence between relative Gromov-Witten invariants of a pair (X, D) and orbifold Gromov-Witten invariants of the r-th root stack Xr along a smooth divisor D. For genus 0 and sufficiently large divisible r, the invariants coincide exactly, showing that orbifold invariants stabilize to relative invariants even when not enumerative. The proof uses deformation theory, virtual fundamental classes, and a comparison of obstruction theories via Costello's theorem on local complete intersections.

ABSTRACT

We prove that genus zero Gromov--Witten invariants of a smooth scheme relative to a smooth divisor coincide with genus zero orbifold Gromov--Witten invariants of an appropriate root stack construction along the divisor.

Motivation & Objective

• To resolve the long-standing question of whether orbifold Gromov-Witten invariants of root stacks stabilize to relative invariants of the pair (X, D).
• To provide a rigorous comparison between the relative and orbifold Gromov-Witten theories in genus 0.
• To clarify the conditions under which orbifold invariants yield the same counts as relative invariants, even when neither is enumerative.
• To extend the understanding of Gromov-Witten invariants beyond the enumerative regime by comparing two distinct geometric constructions: relative maps and orbifold maps.
• To demonstrate that higher-genus invariants do not stabilize, thus showing the genus-0 coincidence is a special phenomenon.

Proposed method

• Constructs the moduli space of relative stable maps to (X, D) and the moduli space of orbifold stable maps to the r-th root stack Xr.
• Uses the virtual fundamental class formalism from Li and Behrend–Fantechi to define invariants in both settings.
• Applies Costello's theorem on virtual fundamental classes via a cartesian diagram of moduli stacks to compare obstruction theories.
• Analyzes the obstruction theory of the relative and orbifold moduli spaces using the Behrend–Fantechi formalism and shows isomorphism of obstruction groups.
• Reduces the comparison to a local complete intersection setting by restricting to dense open substacks of totally non-degenerate objects.
• Employs the universal case of line bundles with sections and expansions to construct a base for the comparison.

Experimental results

Research questions

• RQ1 Do orbifold Gromov-Witten invariants of the r-th root stack Xr stabilize to relative invariants of (X, D) as r → ∞?
• RQ2 Under what conditions do orbifold invariants coincide with relative invariants, even when neither is enumerative?
• RQ3 Why does the genus-0 coincidence fail in higher genera?
• RQ4 Can the obstruction theories of relative and orbifold moduli spaces be compared via a cartesian diagram of stacks?
• RQ5 Is the virtual fundamental class of the relative moduli space isomorphic to that of the orbifold moduli space under the same combinatorial data?

Key findings

• For genus 0 and sufficiently large divisible r, the relative Gromov-Witten invariants of (X, D) and the orbifold invariants of Xr coincide exactly.
• The invariants are equal even when neither is enumerative, demonstrating a deep structural equivalence beyond numerical coincidence.
• The proof relies on showing that the obstruction theories of the relative and orbifold moduli spaces are isomorphic via a cartesian diagram of stacks.
• The comparison is valid only in genus 0; higher-genus orbifold invariants do not stabilize and thus do not coincide with relative invariants.
• The key technical step is the application of Costello’s theorem to a birational, Deligne–Mumford morphism between moduli stacks of stable maps.
• The virtual fundamental classes of the relative and orbifold moduli spaces are isomorphic under the same combinatorial data, confirming the invariance of the invariants.