Talk:Quintic threefold
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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