A mirror theorem for multi-root stacks and applications

TL;DR

This paper establishes a mirror theorem for multi-root stacks derived from a smooth projective variety with a normal crossing divisor, introducing an I-function and demonstrating applications in invariants stabilization, conjecture formulation, and period correspondence.

Contribution

It constructs an I-function for multi-root stacks and applies it to prove invariants stabilization, a conjecture, and period equivalences in mirror symmetry.

Findings

Genus zero invariants stabilize for large root multiplicities.

A version of the local-log-orbifold principle is proven.

Regularized quantum periods match classical periods of mirror potentials.

Abstract

Given a smooth projective variety $X$ with a simple normal crossing divisor $D:=D_1+D_2+...+D_n$, where $D_i\subset X$ are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $X_{D,\vec r}$ by constructing an $I$-function, 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 $X_{D,\vec r}$ stabilize for sufficiently large $\vec r$. (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 $X_{D,\vec r}$.