2004/12/30 by Christian van Enckevort, van Enckevort, Christian, Duco van Straten +1
Mathematics · Physics and Astronomy · #14J32 (Primary) 32S40 #81T30 (Secondary) #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #Number Theory (math.NT) #math.AG #math.NT #msc:14J32 #msc:32S40 #msc:81T30
paper · pdf · doi:10.48550/arxiv.math/0412539
24 pages, 1 figure. Added computation of Euler characteristic and made some minor corrections
openalex publication_date 2004/12/30 · arxiv created 2005/03/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper contains a preliminary study of the monodromy of certain fourth order differential equations, that were called of Calabi-Yau type in math.NT/0402386. Some of these equations can be interpreted as the Picard-Fuchs equations of a Calabi-Yau manifold with one complex modulus, which links up the observed integrality to the conjectured integrality of the Gopakumar-Vafa invariants. A natural question is if in the other cases such a geometrical interpretation is also possible. Our investigations of the monodromies are intended as a first step in answering this question. We use a numerical approach combined with some ideas from homological mirror symmetry to determine the monodromy for some further one-parameter models. Furthermore, we present a conjectural identification of the Picard-Fuchs equation for 5 new examples from Borcea's list and one constructed by Tonoli and conjecture the existence of some new Calabi-Yau three folds. The paper does not contain any theorems or proofs but is, we think, nevertheless of interest.