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Talk:Quintic threefold

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Article rewrite

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Definition

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  • Recall definition of CY manifold
  • Look at defined by a section of
  • Use the adjunction formula to show , hence a quintic 3-fold is given by a smooth degree 5 homogeneous polynomial
  • Show the smoothness condition using the jacobian condition

Hodge numbers

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  • Compute the hodge diamond using Griffiths residues
  • Could also use combinatorial formulas from sheaf cohomology

Deformation theory

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  • relate to the space of deformations

Mirror quintic

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  • page 17 of Cox Katz for quotient
  • remark how taking quotient of just which give a non-trivial stabilizer everywhere, hence a non-trivial orbifold structure, hence that's quotiented out
  • Take the quotient variety
  • Remark on the singularities
  • Use Hodge theory/ cohomology of blow-ups to show the new hodge diamond is a mirror

Picard-Fuchs

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  • Construction of the mirror creates a variation of hodge structures
  • This has a Gauss-Manin connection
  • It's relations are given by the Picard-Fuchs equations

Mirror quintic A-model and B-model

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  • Use Picard-Fuchs to construction correlation function
  • Express it explicitly
  • Give a chart with the first few numbers
  • Show how the first non-trivial number is the number of lines on the quintic

Motivic interpretation

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This page should include a motivic interpretation/ what this mirror symmetry construction means in the motivic world

Schubert calculus

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  • Discuss how to compute the lines on the quintic using schubert calculus