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

Swantje Gährs

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arXiv · 1109.3462

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

Swantje Gährs

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Abstract

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 Poincaré series and the monodromy in the space of period integrals.

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BibTeX RIS

Swantje Gährs. 2011-09-15. Picard-Fuchs equations of special one-parameter families of invertible polynomials. https://arxiv.org/abs/1109.3462

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