Talk:Quintic threefold

Definition

 * Recall definition of CY manifold
 * Look at $$X \subset \mathbb{P}^4$$ defined by a section of $$f \in \Gamma(\mathbb{P}^4, \mathcal{O}(d))$$
 * Use the adjunction formula to show $$d = 5$$, hence a quintic 3-fold is given by a smooth degree 5 homogeneous polynomial
 * Show the smoothness condition using the jacobian condition

Hodge numbers

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

Deformation theory

 * relate $$h^{2,1}$$ to the space of deformations

Mirror quintic

 * page 17 of Cox Katz for quotient
 * remark how taking quotient of just $$\prod \mathbb{Z}/5$$ 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

 * 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

 * Use Picard-Fuchs to construction correlation function $$\langle H, H, H, \rangle$$
 * 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
This page should include a motivic interpretation/ what this mirror symmetry construction means in the motivic world

Schubert calculus

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