Large-root stabilization and polynomiality conjecture for orbifold Gromov–Witten invariants

Let $X$ be a smooth projective variety over $\a9\mathbb{C}$, let $D_1,\dots,D_n$ be smooth, irreducible, nef divisors, and let $r_1,\dots,r_n$ be pairwise coprime natural numbers. Write

$$X_{D,\vec r}:=X_{(D_1,r_1),(D_2,r_2),\ldots,(D_n,r_n)}.$$

Large-root stabilization and polynomiality conjecture. Genus-zero orbifold Gromov–Witten invariants of $X_{D,\vec r}$, after multiplication by suitable powers of the $r_i$, stabilize when the $r_i$ are sufficiently large. Moreover, higher-genus orbifold Gromov–Witten invariants of $X_{D,\vec r}$, after multiplication by suitable powers of the $r_i$, are polynomials in the $r_i$ of degree bounded by $2g-1$ when the $r_i$ are sufficiently large.

The mirror theorem proves stabilization for the relevant genus-zero invariants appearing in the $J$-function, while the stated higher-genus polynomiality is a broader prediction.