Hyperelliptic integrals and mirrors of the Johnson–Kollár del Pezzo surfaces

Corti, Alessio; Gugiatti, Giulia

doi:10.1090/tran/8465

Hyperelliptic integrals and mirrors of the Johnson–Kollár del Pezzo surfaces
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by Alessio Corti and Giulia Gugiatti PDF

Trans. Amer. Math. Soc. 374 (2021), 8603-8637 Request permission

Abstract:

For all integers $k>0$, we prove that the hypergeometric function \[ \widehat {I}_k(\alpha )=\sum _{j=0}^\infty \frac {\bigl ((8k+4)j\bigr )!j!}{(2j)!\bigl ((2k+1)j\bigr )!^2 \bigl ((4k+1)j\bigr )!} \ \alpha ^j \] is a period of a pencil of curves of genus $3k+1$. We prove that the function $\widehat {I}_k$ is a generating function of Gromov–Witten invariants of the family of anticanonical del Pezzo hypersurfaces $X=X_{8k+4} \subset \mathbb {P}(2,2k+1,2k+1,4k+1)$. Thus, the pencil is a Landau–Ginzburg mirror of the family. The surfaces $X$ were first constructed by Johnson and Kollár. The feature of these surfaces that makes our mirror construction especially interesting is that $|-K_X|=|\mathcal {O}_X (1)|=\varnothing$. This means that there is no way to form a Calabi–Yau pair $(X,D)$ out of $X$ and hence there is no known mirror construction for $X$ other than the one given here. We also discuss the connection between our construction and work of Beukers, Cohen and Mellit on hypergeometric functions.

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