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Arithmetic of Degenerating Principal Variations of Hodge Structure: Examples Arising from Mirror Symmetry and Middle Convolution

  Published:2016-01-26
 Printed: Apr 2016
  • Genival da Silva Jr.,
    Department of Mathematics, Campus Box 1146, Washington University in St. Louis, St. Louis, MO, USA
  • Matt Kerr,
    Department of Mathematics, Campus Box 1146, Washington University in St. Louis, St. Louis, MO, USA
  • Gregory Pearlstein,
    Mathematics Department, Mail stop 3368, Texas A$\&$M University, College Station, TX 77843, USA
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Abstract

We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a result of Iritani implies this conjecture for a collection of hypergeometric Calabi-Yau threefold examples studied by Doran and Morgan, the authors investigate a sequence of (non-hypergeometric) examples in dimensions $1\leq d\leq6$ arising from Katz's theory of the middle convolution. A crucial role is played by the Mumford-Tate group (which is $G_{2}$) of the family of 6-folds, and the theory of boundary components of Mumford-Tate domains.
Keywords: variation of Hodge structure, limiting mixed Hodge structure, Calabi-Yau variety, middle convolution, Mumford-Tate group variation of Hodge structure, limiting mixed Hodge structure, Calabi-Yau variety, middle convolution, Mumford-Tate group
MSC Classifications: 14D07, 14M17, 17B45, 20G99, 32M10, 32G20 show english descriptions Variation of Hodge structures [See also 32G20]
Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15]
Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]
None of the above, but in this section
Homogeneous complex manifolds [See also 14M17, 57T15]
Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]
14D07 - Variation of Hodge structures [See also 32G20]
14M17 - Homogeneous spaces and generalizations [See also 32M10, 53C30, 57T15]
17B45 - Lie algebras of linear algebraic groups [See also 14Lxx and 20Gxx]
20G99 - None of the above, but in this section
32M10 - Homogeneous complex manifolds [See also 14M17, 57T15]
32G20 - Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]
 

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