Rational homotopy theory and deformation problems from algebraic geometry. (English) Zbl 0761.32011

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 549-558 (1991).
[For the entire collection see Zbl 0741.00019.]
The author reports on his recent results on how to apply the methods and ideas of rational homotopy theory as developed by Chen, Quillen and Sullivan, to deformations of certain analytic structures. Starting with the motivation initiated by Deligne to control deformation theory by differential graded Lie algebras and quasi-isomorphisms, the author introduces the concept of a formal differential graded algebra, which is, by definition, quasi-isomorphic to one with 0 differentials. It is expected that the deformation theory on an analytic or geometric object is related to some differential graded Lie algebra like in the case of deformations of compact complex manifolds. First, the author states the Comparison Theorem which says, if there is a homomorphism of a differential graded algebra to another, and if it induces an isomorphism on the first cohomology and an injection on the second cohomology, then the corresponding deformation theories are essentially the same. The reader is referred to W. M. Goldman and the author [Ill. J. Math. 34, No. 2, 337-367 (1990; Zbl 0707.32004)]. This (unconsciously) formalizes the idea used by the reviewer in Math. Ann. 222, 275-282 (1976; Zbl 0334.32021)]. Secondly, based on W. M. Goldman and the author [Publ. Math., Inst. Hautes Etud. Sci. 67, 43-96 (1988; Zbl 0678.53059)], the relation with the representation of the fundamental group is discussed. It is also remarked how to interpret, in this context, the result of Bogomolov, proved by Tian [in S. T. Yau (ed.), Mathematical aspects of string theory (1987; Zbl 0651.00012), pp. 629-646] and independently by A. N. Todorov, to the effect that any compact Kähler manifold with trivial canonical bundle (e.g. K3, Calabi- Yau) is unobstructed.
The third topic is a study of the relation between deformations of CR structures as originated by Kuranishi, and those of isolated singularities, in terms of the Comparison Theorem. It is proved that if \((V,0)\) is a normal isolated singularity of \(\dim\geq 4\) and depth \(\geq 3\), then the space of versal family is isomorphic to that of the corresponding CR structure. Here the reader is referred to a forthcoming paper by R. O. Buchweitz and the author. It seems that their proof depend on the previous results of T.Akahori [Invent. Math. 63, 311-344 (1981; Zbl 0496.32015) and ibid. 68, 317-352 (1982; Zbl 0575.32021)] and K. Miyajima [Trans. Am. Math. Soc. 277, 163-172 (1983; Zbl 0525.32019)].
Reviewer: E.Horikawa (Tokyo)


32G07 Deformations of special (e.g., CR) structures
14D15 Formal methods and deformations in algebraic geometry
17B70 Graded Lie (super)algebras
32S05 Local complex singularities