Hypergeometric functions and new mirrors of Fano varieties
File(s)
Author(s)
Gugiatti, Giulia
Type
Thesis
Abstract
In this thesis we outline a strategy to write the hypergeometric function of a Fano weighted complete intersection X of dimension n as a period of a pencil of (n-1)-dimensional varieties. Conjecturally this function is a specialisation of the quantum period of X, thus the ideas in this thesis can be viewed as the basis of a new method to find Landau--Ginzburg mirrors for all Fano weighted complete intersections.
Our strategy builds upon the work of Beukers, Cohen, and Mellit on finite hypergeometric functions, suggesting that the hypergeometric variation attached to X is linked to the variation of intermediate cohomology of a pencil of (n-1+2c)-dimensional varieties, where c is the codimension of X. This pencil can be viewed as a mirror of X of the wrong dimension.
We focus on anticanonical del Pezzo weighted complete intersections. The feature which makes these surfaces especially interesting is that, if the ambient space has all weights greater than 1, they have empty anticanonical linear system, and therefore fall out of the range of the known mirror constructions.
For each positive integer k we exhibit the hypergeometric function of the family of surfaces of degree 8k+4 in P(2,2k+1,2k+1,4k+1) as a period of a pencil of curves of genus 3k+1. This series is the main piece of the Johnson-Kollár classification of anticanonical del Pezzo weighted hypersurfaces. For all k our pencil is a mirror of the family. We build a candidate mirror for the family of surfaces of degree 15 in P(3,3,5,5) by identifying the variation of H^3 of its wrong dimension mirror with a quotient of the variation of H^1 of a pencil of genus-six curves.
We also provide evidence that our constructions are compatible with the work of Hori-Vafa.
We conclude by discussing our ideas to prove the conjectural Hodge--theoretic translation of the work of Beukers, Cohen, and Mellit.
Our strategy builds upon the work of Beukers, Cohen, and Mellit on finite hypergeometric functions, suggesting that the hypergeometric variation attached to X is linked to the variation of intermediate cohomology of a pencil of (n-1+2c)-dimensional varieties, where c is the codimension of X. This pencil can be viewed as a mirror of X of the wrong dimension.
We focus on anticanonical del Pezzo weighted complete intersections. The feature which makes these surfaces especially interesting is that, if the ambient space has all weights greater than 1, they have empty anticanonical linear system, and therefore fall out of the range of the known mirror constructions.
For each positive integer k we exhibit the hypergeometric function of the family of surfaces of degree 8k+4 in P(2,2k+1,2k+1,4k+1) as a period of a pencil of curves of genus 3k+1. This series is the main piece of the Johnson-Kollár classification of anticanonical del Pezzo weighted hypersurfaces. For all k our pencil is a mirror of the family. We build a candidate mirror for the family of surfaces of degree 15 in P(3,3,5,5) by identifying the variation of H^3 of its wrong dimension mirror with a quotient of the variation of H^1 of a pencil of genus-six curves.
We also provide evidence that our constructions are compatible with the work of Hori-Vafa.
We conclude by discussing our ideas to prove the conjectural Hodge--theoretic translation of the work of Beukers, Cohen, and Mellit.
Version
Open Access
Date Issued
2020-08
Date Awarded
2021-01
Copyright Statement
Creative Commons Attribution NonCommercial NoDerivatives Licence
Advisor
Corti, Alessio
Sponsor
EPSRC Centre for Doctoral Training in Geometry and Number Theory (LSGNT)
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)