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, where D_isubset X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks X_{D,{overrightarrow{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}} stabilize for sufficiently large overrightarrow{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}}.