The Picard-Fuchs equation and its monodromy for a family of Calabi-Yau hypersurfaces in CP(N-1)

We derive the Picard-Fuchs (P-F) equation, with no restrictions on the dimensions, for a one-parameter family of Calabi-Yau hypersurfaces in CPN-1 , in two different ways. One is algebraic and it is based on David Morrison's approach in [20], which uses Phillip Griffiths' results regarding the reduction of pole order for rational differential forms in Projective space. The other way is geometric and it is based on Leray's residues and also it is closer to the original approach of Candelas et al. in [6]. We prove the equivalence of the two results obtained from using the two different approaches. In general, the P-F equation is a meromorphic differential equation with regular singular points and it represents a flat connection on a cohomology bundle over CP1 , for a one-parameter deformation. The P-F equation encodes the deformation of complex structure for the Mirror family of the family of hypersurfaces in Projective space.; Inspired by the methods developed by Donald Babbitt and V.S.Varadarajan in [2), regarding the formal reduction theory of meromorphic differential equations with regular singularities, we also investigate the local monodromies, not just the global monodromy generators, which is the case in the paper by Candelas et al. [6]. Using these methods, we reduce our matrix P-F equation to a canonical form at each singular point. This gives us all the local monodromy matrices for the P-F equation at the singular points. We then explicitly verify the Monodromy theorem, proved in the abstract setting by Brieskorn, Borel and others, using heavy machinery from algebraic geometry, which states that all the local monodromy eigenvalues are roots of unity. In particular, we show that, for our family of cases, all the local monodromy eigenvalues are square roots of unity.; We derive the monodromy generators of the global Monodromy group using Barnes' integral representation of the generalized hypergeometric function to describe the analytic continuation of a basis of solutions for the P-F equation, an approach based on the paper of Candelas et al., [6].