A mirror theorem for multi-root stacks and applications

Abstract

Let X be a smooth projective variety with a simple normal crossing divisor $$D:=D_1+D_2+\cdots +D_n$$ D : = D 1 + D 2 + ⋯ + D n , where $$D_i\subset X$$ D i ⊂ X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $$X_{D,{\overrightarrow{r}}}$$ X D , r → by constructing an I-function lying in a slice of Givental’s Lagrangian cone for Gromov–Witten theory of multi-root stacks. We provide three applications: (1) We show that some genus zero invariants of $$X_{D,\overrightarrow{r}}$$ X D , r → stabilize for sufficiently large $$\overrightarrow{r}$$ r → . (2) We state a generalized local-log-orbifold principle conjecture and prove a version of it. (3) We show that regularized quantum periods of Fano varieties coincide with classical periods of the mirror Landau–Ginzburg potentials using orbifold invariants of $$X_{D,\overrightarrow{r}}$$ X D , r → .

The Local-Orbifold Correspondence for Simple Normal Crossing Pairs

“…We study the relationship between the genus 0 local Gromov-Witten theory of ⊕ n i=1 O X (−D i ) and the genus 0 orbifold Gromov-Witten theory of the multiroot stack X D, r . Our main result is a positive answer to [27,Conjecture 1.8…”

“…The case follows by combining the smooth divisor local-logarithmic correspondence [29] in its strong form (see [13, Introduction] or [27, equation (2)]), together with the smooth divisor logarithmic-orbifold correspondence [1, 28].…”

“…The smooth divisor case of Theorem A follows by combining the orbifold-logarithmic correspondence [1, 28] with the strong form [13, 27] of the local-logarithmic correspondence [29]. Some cases of Theorem A for normal crossing divisors were numerically verified in [27, §5.2], by computing the J -functions of both sides.…”