This Lecture Notes volume, a sequel to volumes 110S and 1214, presents to English speaking readers further two issues of the Vo- ronezh University Press series "Novoe v global'nom analiza" (New Developments in Global Analysis). These issues, published in Russian in March, 1987 and in November, 1987, are entitled "Global Analysis and Mathematical Physics" and "Geometry and Theory of S.ingularities in Nonlinear Equations", respectively. For this translation volume we have selected the articles of survey (expository) character and papers giving detailed accounts of research results. Short commu- nications and the survey article of Yu.P.Solov'ev (March, 1987) are omitted here (the material of the latter will be published in Eng- lish elsewhere).

The present. volume is the second volume of the book "Singularities of Differentiable Maps" by V.1. Arnold, A. N. Varchenko and S. M. Gusein-Zade. The first volume, subtitled "Classification of critical points, caustics and wave fronts", was published by Moscow, "Nauka", in 1982. It will be referred to in this text simply as "Volume 1". Whilst the first volume contained the zoology of differentiable maps, that is it was devoted to a description of what, where and how singularities could be encountered, this volume contains the elements of the anatomy and physiology of singularities of differentiable functions. This means that the questions considered in it are about the structure of singularities and how they function. Another distinctive feature of the present volume is that we take a hard look at questions for which it is important to work in the complex domain, where the first volume was devoted to themes for which, on the whole, it was not important which field (real or complex) we were considering. Such topics as, for example, decomposition of singularities, the connection between singularities and Lie algebras and the asymptotic behaviour of different integrals depending on parameters become clearer in the complex domain. The book consists of three parts. In the first part we consider the topological structure of isolated critical points of holomorphic functions. We describe the fundamental topological characteristics of such critical points: vanishing cycles, distinguished bases, intersection matrices, monodromy groups, the variation operator and their interconnections and method of calculation.

Foliation theory has its origins in the global analysis of solutions of ordinary differential equations: on an n-dimensional manifold M, an [autonomous] differential equation is defined by a vector field X ; if this vector field has no singularities, then its trajectories form a par­ tition of M into curves, i.e. a foliation of codimension n - 1. More generally, a foliation F of codimension q on M corresponds to a partition of M into immersed submanifolds [the leaves] of dimension ,––––,- - . - – p = n - q. The first global image that comes to mind is 1––––;- - - - - - that of a stack of "plaques". 1––––-;- - - - - - Viewed laterally [transver­ 1––––1- - - – sally], the leaves of such a 1––––1 - - - - -. stacking are the points of a 1––––1–- ––. quotient manifold W of di­ L….. -' _ mension q. ––-~) W M Actually, this image corresponds to an elementary type of folia­ tion, that one says is "simple". For an arbitrary foliation, it is only l- u L ally [on a "simpIe" open set U] that the foliation appears as a stack of plaques and admits a local quotient manifold. Globally, a leaf L may - - return and cut a simple open set U in several plaques, sometimes even an infinite number of plaques.

These notes present very recent results on compact Kähler-Einstein manifolds of positive scalar curvature. A central role is played here by a Lie algebra character of the complex Lie algebra consisting of all holomorphic vector fields, which can be intrinsically defined on any compact complex manifold and becomes an obstruction to the existence of a Kähler-Einstein metric. Recent results concerning this character are collected here, dealing with its origin, generalizations, sufficiency for the existence of a Kähler-Einstein metric and lifting to a group character. Other related topics such as extremal Kähler metrics studied by Calabi and others and the existence results of Tian and Yau are also reviewed. As the rudiments of Kählerian geometry and Chern-Simons theory are presented in full detail, these notes are accessible to graduate students as well as to specialists of the subject.