Periods and algebraic deRham cohomology

Abstract

It is known that the algebraic \deRham cohomology group $\hDR{i}(X_0/\Q)$ of a nonsingular variety $X_0/\Q$ has the same rank as the rational singular cohomology group $\h^i\sing(\Xh;\Q)$ of the complex manifold $\Xh$ associated to the base change $X_0\times_{\Q}\C$. However, we do not have a natural isomorphism $\hDR{i}(X_0/\Q)\iso\h^i\sing(\Xh;\Q)$. Any choice of such an isomorphism produces certain integrals, so called periods, which reveal valuable information about $X_0$. 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 $\zeta(2)$ as a period and a variation of mixed Hodge structures used by Goncharov.