Relative and orbifold Gromov�Witten invariants

Abstract

We prove that genus 0 Gromov-Witten invariants of a smooth scheme relative to a smooth divisor coincide with genus 0 orbifold Gromov-Witten invariants of an appropriate root stack construction along the divisor. The proof is given at the level of virtual fundamental classes.

Gromov–Witten theory of complete intersections via nodal invariants

“…We review here the construction of the virtual class in relative Gromov–Witten theory based on working relatively to the moduli space of maps to a universal target. For details we refer to [2, Section 5; 3, Section 3.2].…”

“…For example, Graber–Vakil gave a more compact description of the virtual class by defining a perfect obstruction theory relative to $${\mathfrak {M}}_G \times \mathcal {T}$$, where $$\mathcal {T}$$ is the stack of expanded degenerations [33, Section 2.8]. More recently, Abramovich–Cadman–Wise have provided an alternative description obtained by working relatively to the space of maps to a universal target [2, Section 5]; see also [3, Section 3.2]. We follow here the Abramovich–Cadman–Wise approach which is technically simpler.…”

“…Let $$\mathfrak {P}_\Gamma (\mathcal {A},\mathcal {D})$$ be the substack of $$\Gamma$$‐marked prestable maps to $$(\mathcal {A},\mathcal {D})$$ whose $$\Gamma$$‐marked nodes are not distinguished. The latter requirement on nodes ensures that the proof of [2, Lemma 4.1.2] applies to $$\mathfrak {P}_\Gamma (\mathcal {A},\mathcal {D})$$. Hence, $$\mathfrak {P}_\Gamma (\mathcal {A},\mathcal {D})$$ is equidimensional and has an ordinary fundamental class $$[\mathfrak {P}_\Gamma (\mathcal {A},\mathcal {D})]$$.…”

“…The space (𝑋, 𝐷) 2 and the natural 𝑛-pointed generalization (𝑋, 𝐷) 𝑛 have previously appeared in relative Gromov-Witten theory in the formulation of the Gromov-Witten/Pairs correspondence for relative theories [70, Section 1.2].…”

“…The natural morphism (3.1) induces a morphism $$$$\begin{equation*} \nonumber \eta _G \colon \overline{\mathcal {M}}_G(X,D) \rightarrow {\mathfrak {M}}_G(\mathcal {A},\mathcal {D}). \end{equation*}$$$$By [2, Section 5], the virtual class in relative Gromov–Witten theory can be defined using the perfect obstruction theory relative to $$\eta _G$$ given by $$$$\begin{equation} (R\pi _{*}f^{*}T_{(X,D)})^\vee , \end{equation}$$$$where $$\pi \colon C \rightarrow \overline{\mathcal {M}}_G(X,D)$$ is the universal curve, $$f \colon C \rightarrow \mathcal {X} \rightarrow X$$ is the composition of the universal relative stable map with the contraction of the expanded target on $$X$$, and $$T_{(X,D)}$$ is the log tangent bundle of the pair $$(X,D)$$. The virtual class is then defined by …”