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Deformations of highly symmetric Calabi-Yau Grassmannian hypersurfaces

Adriana Salerno, Ursula Whitcher, Chenglong Yu

Abstract

We use arithmetic and Hodge-theoretic techniques to study pencils of Calabi-Yau varieties realized as highly symmetric hypersurfaces in Grassmannians and their quotients, demonstrating that their geometric properties are distinct from the classical mirrors of Calabi-Yau Grassmannian hypersurfaces.

Deformations of highly symmetric Calabi-Yau Grassmannian hypersurfaces

Abstract

We use arithmetic and Hodge-theoretic techniques to study pencils of Calabi-Yau varieties realized as highly symmetric hypersurfaces in Grassmannians and their quotients, demonstrating that their geometric properties are distinct from the classical mirrors of Calabi-Yau Grassmannian hypersurfaces.
Paper Structure (11 sections, 4 theorems, 20 equations, 3 tables)

This paper contains 11 sections, 4 theorems, 20 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Mirror constructions and Calabi-Yau pencils
  3. The Greene--Plesser mirror quintic
  4. Classical mirrors of Calabi-Yau hypersurfaces in Grassmannians
  5. Arrow pencils and a candidate mirror
  6. The arrow pencil in $G(2,4)$
  7. Finite field data and the Hasse-Witt invariant
  8. Griffiths ring for complete intersections
  9. Middle cohomology of arrow pencils
  10. Griffiths ring for Grassmannian hypersurfaces
  11. Arrow pencils and arrow pencil alternatives

Key Result

Proposition 3.3

No truncation relationship of the form described in Hypothesis hyp:hypergeometricTruncation holds for $p=5, 7$, or $11$.

Theorems & Definitions (21)

  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Conjecture 2.8: CDK
  • Example 3.2
  • Proposition 3.3
  • ...and 11 more