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Homology supported in Lagrangian submanifolds in mirror quintic threefolds

Part of: Symplectic geometry, contact geometry Surfaces and higher-dimensional varieties Families, fibrations

Published online by Cambridge University Press:  11 September 2020

Daniel López Garcia
Daniel López Garcia*
Affiliation:
Instituto de Matematica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, Rio de Janeiro 22460-320, Brazil
*
e-mail: daflopez@impa.br
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Abstract

In this note, we study homology classes in the mirror quintic Calabi–Yau threefold that can be realized by special Lagrangian submanifolds. We have used Picard–Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers $p,$ we can compute the orbit modulo p. We conjecture that the orbit in homology with coefficients in $\mathbb {Z}$ can be determined by these orbits with coefficients in $\mathbb {Z}_p$ .

Keywords

Lagrangian spheresmirror quintic threefoldmonodromy actionLefschetz fibrationvanishing cycles

MSC classification

Primary: 14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)
Secondary: 14J30: $3$-folds 14J33: Mirror symmetry 53D05: Symplectic manifolds, general 53D12: Lagrangian submanifolds; Maslov index
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 3 , September 2021 , pp. 709 - 724
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Copyright
© Canadian Mathematical Society 2020

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