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Hypergeometric local systems over $\mathbb{Q}$ with Hodge vector $(1,1,1,1)$

Giulia Gugiatti, Fernando Rodriguez Villegas

Abstract

We consider all irreducible rank-4 hypergeometric local systems defined over $\mathbb{Q}$ that support a rational one-dimensional variation of Hodge structures of weight 3 and Hodge vector $(1,1,1,1)$. Up to a natural equivalence there are only 47 cases. The first 14 cases have maximally unipotent monodromy at one point and have been extensively studied in the literature. We show that all 47 local systems are associated to families of generically smooth threefolds and we analyze the geometry and arithmetic at their conifold point.

Hypergeometric local systems over $\mathbb{Q}$ with Hodge vector $(1,1,1,1)$

Abstract

We consider all irreducible rank-4 hypergeometric local systems defined over that support a rational one-dimensional variation of Hodge structures of weight 3 and Hodge vector . Up to a natural equivalence there are only 47 cases. The first 14 cases have maximally unipotent monodromy at one point and have been extensively studied in the literature. We show that all 47 local systems are associated to families of generically smooth threefolds and we analyze the geometry and arithmetic at their conifold point.
Paper Structure (21 sections, 56 equations, 2 tables)

This paper contains 21 sections, 56 equations, 2 tables.

Table of Contents

  1. Goal 1.
  2. Goal 2.
  3. Related and future work
  4. Acknowledgments
  5. Geometric realisation
  6. Family parameters and gamma vectors
  7. Canonical pairs and toric models
  8. Dimension reduction
  9. Union of gamma vectors with relatively prime entries
  10. The pair $(Y,w)$.
  11. Connection with $(Z, \pi)$.
  12. Generalisations of $\S\ref{['sec:gcd=1']}$
  13. Application (2): non-relatively prime entries.
  14. Application (3): union of arbitrary gamma vectors.
  15. The geometry of total twists
  16. ...and 6 more sections

Theorems & Definitions (5)

  • Definition 1
  • Example 2
  • Remark 3
  • Remark 4
  • Remark 5