Picard-Fuchs equations of special one-parameter families of invertible polynomials

Details Picard-Fuchs equations of special one-parameter families of invertible polynomials Picard-Fuchs equations of special one-parameter families of invertible polynomials

2011-09-15

arxiv.org/abs/1109.3462v1

Mathematics

Algebraic Geometry

Ph.D. Thesis, 103 pages, 29 figures

Scientific paper

The thesis deals with calculating the Picard-Fuchs equation of special one-parameter families of invertible polynomials. In particular, for an invertible polynomial $g(x_1,...,x_n)$ we consider the family $f(x_1,...,x_n)=g(x_1,...,x_n)+s\cdot\prod x_i$, where $s$ denotes the parameter. For the families of hypersurfaces defined by these polynomials, we compute the Picard-Fuchs equation, i.e. the ordinary differential equation which solutions are exactly the period integrals. For the proof of the exact appearance of the Picard-Fuchs equation we use a combinatorial version of the Griffiths-Dwork method and the theory of \GKZ systems. As consequences of our work and facts from the literature, we show the relation between the Picard-Fuchs equation, the Poincar\'{e} series and the monodromy in the space of period integrals.

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