The Local-Orbifold Correspondence for Simple Normal Crossing Pairs

Abstract

For X a smooth projective variety and $D=D_1+\dotsb +D_n$ a simple normal crossing divisor, we establish a precise cycle-level correspondence between the genus $0$ local Gromov–Witten theory of the bundle $\oplus _{i=1}^n \mathcal {O}_X(-D_i)$ and the maximal contact Gromov–Witten theory of the multiroot stack $X_{D,\vec r}$ . The proof is an implementation of the rank-reduction strategy. We use this point of view to clarify the relationship between logarithmic and orbifold invariants.

Stable maps to Looijenga pairs: orbifold examples

“…In the case of an irreducible smooth nef divisor, the correspondence between genus 0 log and local GW invariants was proven in all dimensions at the cycle-level in [36], with various extensions in [8-10, 14, 15, 23-25, 34, 55, 61, 62]. The naive conjectural extension of this log-local correspondence at the cycle level for normal crossings divisors has been recently disproved [8,55]. However, the numerical version of the log-local correspondence for normal crossing divisors seems to hold in a number of cases of great interest: for example this was proved for point insertions of orbifold toric pairs in [14], and for point invariants of log Calabi-Yau surfaces with nef D i in [15].…”

“…2 As discussed in more details in [15, §1.4], point insertions and the log Calabi-Yau condition are both crucial assumptions allowing us to obtain the numerical log-local correspondence despite the general negative results of [8,55]. In [15, §5], we gave a conceptual proof by degeneration of the numerical log-local correspondence for log Calabi-Yau surfaces with two boundary components.…”