A mirror theorem for multi-root stacks and applications

Abstract

Abstract Let X be a smooth projective variety with a simple normal crossing divisor $$D:=D_1+D_2+\cdots +D_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where $$D_i\subset X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> are smooth, irreducible and nef. We prove a mirror theorem for multi-root stacks $$X_{D,{\overrightarrow{r}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mrow> </mml:msub> </mml:math> 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 $$X_{D,\overrightarrow{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mrow> </mml:msub> </mml:math> stabilize for sufficiently large $$\overrightarrow{r}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:math> . (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 $$X_{D,\overrightarrow{r}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>r</mml:mi> <mml:mo>→</mml:mo> </mml:mover> </mml:mrow> </mml:msub> </mml:math> .

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