2005/06/07 by Benjamin Friedrich, Friedrich, Benjamin
Mathematics · #14F40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #math.AG #msc:14F40
paper · pdf · doi:10.48550/arxiv.math/0506113
103 pages, 12 figures, diploma thesis
arxiv created 2005/06/07 · openalex publication_date 2005/06/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is known that the algebraic \deRham cohomology group \hDRi(X0/\Q) of a nonsingular variety X0/\Q has the same rank as the rational singular cohomology group \hi\sing(\Xh;\Q) of the complex manifold \Xh associated to the base change X0×\Q\C. However, we do not have a natural isomorphism \hDRi(X0/\Q)\iso\hi\sing(\Xh;\Q). Any choice of such an isomorphism produces certain integrals, so called periods, which reveal valuable information about X0. The aim of this thesis is to explain these classical facts in detail. Based on an approach of Kontsevich, different definitions of a period are compared and their properties discussed. Finally, the theory is applied to some examples. These examples include a representation of ζ(2) as a period and a variation of mixed Hodge structures used by Goncharov.