Infinitesimal Methods in Hodge Theory

Abstract

This is a set of survey lectures on the (infinitesimal) variation of Hodge structures. The contents are divided into eight lectures, including an introduction to Hodge theory. In this paper, the author concentrates mainly on variation of Hodge structures, the Abel-Jacobi map and normal functions. Lecture 1 is an introdution to Hodge theory. The author introduces Kähler geometry, namely the Lefschetz decomposition, harmonic forms and Hodge decomposition of a compact Kähler manifold. The beginning of Lecture 2 is devoted to the definition of the cycle map. Then one can define the generalization of the Abel-Jacobi map from the kernel $Z^p_h(X)$ of the cycle map to the intermediate Jacobian $J^p(X)$ of $X$. An element in $Z^p_h(X)$ is called a cycle homologically equivalent to $0$. If an algebraic family $\scr Z$ of homologically equivalent to 0 cycles on $\scr X \to T$ is given, then this construction is applied to constructing a section to the family of the intermediate Jacobian $\scr J^p$. This is called the normal function associated to $\scr Z$. This is treated in Lecture 6. There the author introduces Griffiths' infinitesimal invariant. The rest of Lecture 2 is spent on introducing Deligne cohomology. In Lecture 3, he introduces infinitesimal variations of Hodge structure and the Kodaira-Spencer map. Applying the Kodaira-Spencer map successively, he defines the Yukawa coupling (named after the physicist H. Yukawa). Lecture 4 is devoted to a more concrete but important example, hypersurfaces in $\bold P^n$. The computation of the infinitesimal variation of hypersurfaces on $\bold P^n$ is reduced to that of the Jacobian ring. Then the author mentions the infinitesimal Torelli theorem. In Lecture 5, he introduces the Hodge theory of open varieties. Hodge theory is naturally generalized to open varieties within the framework of mixed Hodge structures given by Deligne. Here one can find an account of the era in which this notion was originated. Lecture 7 is devoted to the generic Torelli theorem for hypersurfaces. To recover the equations of hypersurfaces from the infinitesimal data, one must investigate the symmetrizer lemma. In Lecture 8, the author mentions his recent original work, which is mainly concerned with Nori's connectedness theorem. He gives some results on the image of the map from the Chow ring to Deligne cohomology.