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Minimal log discrepancies of hypersurface mirrors

Louis Esser

Abstract

For certain quasismooth Calabi-Yau hypersurfaces in weighted projective space, the Berglund-Hübsch-Krawitz (BHK) mirror symmetry construction gives a concrete description of the mirror. We prove that the minimal log discrepancy of the quotient of such a hypersurface by its toric automorphism group is closely related to the weights and degree of the BHK mirror. As an application, we exhibit klt Calabi-Yau varieties with the smallest known minimal log discrepancy. We conjecture that these examples are optimal in every dimension.

Minimal log discrepancies of hypersurface mirrors

Abstract

For certain quasismooth Calabi-Yau hypersurfaces in weighted projective space, the Berglund-Hübsch-Krawitz (BHK) mirror symmetry construction gives a concrete description of the mirror. We prove that the minimal log discrepancy of the quotient of such a hypersurface by its toric automorphism group is closely related to the weights and degree of the BHK mirror. As an application, we exhibit klt Calabi-Yau varieties with the smallest known minimal log discrepancy. We conjecture that these examples are optimal in every dimension.
Paper Structure (7 sections, 12 theorems, 40 equations, 1 figure)

This paper contains 7 sections, 12 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. Minimal log discrepancies and toric stacks
  4. Hypersurfaces in Fake Weighted Projective Stacks
  5. Berglund-Hübsch-Krawitz Mirror Symmetry
  6. Minimal Log Discrepancies of Hypersurface Quotients
  7. Applications

Key Result

Theorem 1.1

Let $V_d \subset \mathbb{P}_{\mathbb{C}}(a_0,\ldots,a_{n+1})$ be a well-formed quasismooth Calabi--Yau hypersurface defined by a polynomial equation of degree $d$ with the same number of monomials as variables such that its matrix of exponents is invertible (also called a Delsarte polynomial). Suppo

Figures (1)

  • Figure 1: The directed graph corresponding to a Delsarte potential function of shape $f = x_1^{b_1}x_4 + x_2^{b_2} + x_3^{b_3}x_6 + x_4^{b_4}x_7 + x_5^{b_5}x_1 + x_6^{b_6} + x_7^{b_7}x_5.$ This potential is composed of three atoms.

Theorems & Definitions (23)

  • Theorem 1.1: \ref{['mainmldthm']}
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 13 more