Rigidity, Past and Present

This note is an elaborated version of the talk with the same title I gave in the January 2006 workshop at the HRI. The talk was aimed at (complex) algebraic geometers with little background in Hodge theory. In the talk, I have tried to give as much details from Hodge theory so as to allow the audience to follow the main arguments. Much of these details are presented here as well. I try to avoid to become too technical by substituting references to the existing literature. The point of departure is a beautiful result of Arakelov [Arak] which is explained below in § 1.2. A brief sketch of the original proof is given in § 1.3. Since its appearance, this theorem motivated many people to look for analogs in the number theoretic setting. Analogs which eventually led Faltings to his proof [Fa83b] of the Mordell conjecture conjecture. Indeed, a version of Arakelov’s theorem for Abelian varieties appears by the same author [Fa83a] in the same journal: it is the article which is just preceding the famous article in which the Mordell conjecture has been proven. In [Pe90] I proved a generalization of Faltings’ result for variations of Hodge structures. At the request of the organizers of the conference I explained this proof and added a few recent developments. Among the latter I mention the classification of non-rigid variations of K3-surfaces [S-Zu] due to M.-H. Saito and S. Zucker, as well as for Abelian varieties [Sa], due to M.-H. Saito. The reader will find a few new proofs of some recent results as well. For example, one of the main results of [L-T-Y-Z]: any family of Calabi-Yau’s with maximally unipotent local monodromy at some point of the boundary must be rigid (Cor 3.5). The same approach gives an easy proof of a result from [VZ]: a variation with maximal Higgs field is rigid (Prop. 3.7). Contrary to when I wrote [Pe90], nowadays some good introductory works to Griffiths’ theory have appeared such as [V] and [C-P-M]. The latter treats the Lie-theoretic background which I need. For that reason I often refer to it for details omitted in the presentation below. At the conference H. Shiga pointed out to me that he together with Y. Imayoshi in [I-S] proved Arakelov’s rigidity result by analytic means.