A mirror theorem for multi-root stacks and applications

Abstract

Given a smooth projective variety X with a simple normal crossing divisor D := D 1 + D 2 + ... + D n , where D i ⊂ X are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks X D, 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, r stabilize for sufficiently large 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, r .

Degenerations, fibrations and higher rank Landau-Ginzburg models

“…For LG models which are mirror to smooth pairs (X, D), the relative periods considered in [DKY19] are mirror to generating functions of genus zero Gromov-Witten invariants of (X, D) by the relative mirror theorem in [FTY19]. Now, if we have higher rank LG models which are mirror to a simple normal crossings pairs (X, D), then the relative periods that we consider in this paper are mirror to genus zero Gromov-Witten invariants of (X, D) ( [TY20a], see also Section 9).…”

“…In this paper, we consider the type of invariants associated to a simple normal crossing pairs defined in [TY20b] which fits well into our context. In [TY20a], a mirror theorem has been proved to relate the formal Gromov-Witten invariants of infinite root stacks, which are invariants associated to simple normal crossing pairs and defined in [TY20b], and periods.…”

“…More generally, a mirror theorem can be stated using Givental's formalism (Givental's Lagrangian cone etc.). A mirror theorem for the formal Gromov-Witten invariants of infinite root stacks is proved in [TY20a] as a limit of the mirror theorem for multi-root stacks.…”

“…In Theorem 1.3 and Theorem 1.4, we glue relative periods of two rank 2 LG models to obtain the relative periods for rank 1 LG models. By the relative mirror theorem of [TY20a], relative periods of higher rank LG models are mirror to the genus zero Gromov-Witten invariants of simple normal crossings pairs defined in [TY20b].…”

The local-orbifold correspondence for simple normal crossings pairs

“…Relation to previous work. The smooth divisor case of Theorem A follows by combining the orbifold-logarithmic correspondence [ACW17,TY20b] with the strong form [FW20,TY20a] of the local-logarithmic correspondence [vGGR19]. Some cases of Theorem A for normal crossings divisors were numerically verified in [TY20a, §5.2], by computing the J-functions of both sides.…”