Talk:Quintic threefold

Article rewrite edit

Definition edit

  • 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 edit

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

Deformation theory edit

  • relate   to the space of deformations

Mirror quintic edit

  • 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 edit

  • 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 edit

  • 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 edit

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

Schubert calculus edit

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