One of the major discoveries of the last two decades of the twentieth century in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the a comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.

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The aim of this address is to give an overview of the main questions and results of the structure theory of higher dimensional algebraic varieties.

This book gives a comprehensive treatment of the singularities that appear in the minimal model program and in the moduli problem for varieties. The study of these singularities and the development of Mori's program have been deeply intertwined. Early work on minimal models relied on detailed study of terminal and canonical singularities but many later results on log terminal singularities were obtained as consequences of the minimal model program. Recent work on the abundance conjecture and on moduli of varieties of general type relies on subtle properties of log canonical singularities and conversely, the sharpest theorems about these singularities use newly developed special cases of the abundance problem. This book untangles these interwoven threads, presenting a self-contained and complete theory of these singularities, including many previously unpublished results.

The central theme of this article is the study of deformations of surface singularities using recent advances in three dimensional geometry. The basic idea is the following. Let X 0 be a surface singularity and consider a one parameter deformation {Xo: teA}. Then the total space X=UX t is a three dimensional object. One can attempt to use the geometry of X to get information about the surface X~.In general X is very singular and so one can try to study it via a suitable resolution of singularities f: X' -~ X. The existence of a resolution was established by Zariski; the problem is that there are too many of them, none particularly simple.Mori and Reid discovered that the best one can hope for is a partial resolution f: X'~ X where X' possesses certain mild singularities but otherwise is a good analog of the minimal resolution of surface singularities.The search for such a resolution is known as Mori's program (see e.g. KMM]). After substantial contributions by several mathematicians (Benveniste, Kawamata, Kollfir, Mori, Reid, Shokurov, Vichweg) A precise formulation of the result we need will be provided at the end of the introduction.In certain situations X0 will impose very strong restrictions on X' and one can use this to obtain information about X and X~ for t 40.The first application is in chapter two. Teissier [Tel posed the following problem. Let {X~ : s~S} be a flat family of surfaces parameterized by the connected space S. Let X s be the minimal resolution of X~. In general {Xs: seS} is not a flat family of surfaces, and it is of interest to find necessary and sufficient conditions for this to hold. If {J~s: s~S} is a flat family, then K 2-is locally constant on S. One can Xs hope that the converse is also true (at least after a finite and surjective base change). This was proved by Laufer [La, 5.7] under the additional assumption that the Xs have isolated Gorenstein singularities. The main result of chapter two (2.10) generalizes this to the case where the X~ have arbitrary isolated singularities. In the presence of nonisolated singularities the converse is usually false.Chapter three provides a new approach to the study of the deformation space of a quotient singularity. The main result (Theorem 3.9) relates the irreducible components of the deformation space to certain partial resolutions of the singularity. This gives an algorithm to compute the number of components of the deformation space and the dimension of the components in terms of the dual graph of the minimal resolution. It is our belief that this method will lead eventually to the understanding of all small deformations of a quotient singularity, but so far our attempts have been frustrated.The next two chapters contain results pertaining to the problem of compactifying the moduli of surfaces. This involves the study of certain singular surfaces that appear as limits of smooth ones. A good class of such singular surfaces is suggested by the above-mentioned results of three dimensional geometry. The possible singularities of such surfaces...

We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and Varchenko, mostly for the study of hypersurface singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but it also takes into account more quantitative informations of great interest for complex analysis and complex differential geometry. We give as an application a new derivation of criteria for the existence of Kähler-Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, 3 new examples of rigid Kähler-Einstein Del Pezzo surfaces with quotient singularities are obtained.Résumé. Nous introduisons les exposants de singularités complexes des fonctions plurisousharmoniques et démontrons un théorème de semi-continuité général pour ceux-ci. Le conceptétudié contient comme cas particulier des concepts voisins qui ontété considérés par exemple par Arnold et Varchenko, principalement pour l'étude des singularités d'hypersurfaces. La version plurisousharmonique repose en définitive sur une réduction au cas algébrique, mais elle prend aussi en compte des informations quantitatives d'un grand intérêt pour l'analyse complexe et la géométrie différentielle complexe. Nous décrivons en application une nouvelle approche des critères d'existence de métriques Kähler-Einstein pour les variétés de Fano, en nous inspirant des idées originales de Nadel -mais avec des simplifications importantes de la technique, une fois que le résultat de semi-continuité est utilisé comme outil de base. Grâceà ces critères, nous obtenons trois nouveaux exemples de surfaces de Del Pezzoà singularités quotients, rigides, possédant une métrique de Kähler-Einstein. 2 J.-P. Demailly, J. Kollár, Semi-continuity of complex singularity exponents Contents

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