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Higher Dimensional Complex Geometry Astérisque HERBERT CLEMENS JÁNOS KOLLÁR SHIGEFUMI MORI Higher dimensional complex geometry, A Summer Seminar at the University of Utah, Slat Lake City, 1987 Astérisque, tome 166 (1988) <http://www.numdam.org/item?id=AST_1988__166__1_0> © Société mathématique de France, 1988, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 166 ASTÉRISQUE 1988 HIGHER DIMENSIONAL COMPLEX GEOMETRY Herbert CLEMENS, János KOLLÁR, Shigefumi MORI A Summer Seminar at the University of Utah, Salt Lake City, 1987 SOCIÉTÉ MATHÉMATIQUE DE FRANCE Publié avec le concours du CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE A.M.S. Subjects Classification : 14-02, 14-C-30, 14-C-99, 14-E-05, 14-E-30, 14-E-35, 14-J-30, 32-J-25, 53-B-35, 53-C-35. Introduction These notes originated a t a seminar that was held during Jul y and August o f 198 7 at Salt Lake City. Th e original aim of the seminar was to get an overview o f the following three topics: 1. Recent advances i n the classification program o f three (an d higher) dimensional algebraic varieties. 2. Existence o f rational curves and other specia l subvarieties. 3. Existence an d nature o f special metrics o n varieties. We also hoped to then go further and study the relationship s between these three approaches. Tim e however proved to be insufficient t o complete eve n the limited program. The firs t part o f the program was considered i n detail. I n that part, the central theme i s the investigatio n o f varieties on which the canonical class i s not numerically effective. Fo r smooth threefolds this was done i n [Ml ] and later extended considerably. Th e original approach o f [Ml ] is geometrically ver y clear, therefore i t i s given i n detail. Subsequen t generalizations were also considered a t length. Considerable attention was paid also to the stud y of specia l curves o n hypersurfaces an d some related examples. Ther e seem s to be a lot of experimental evidenc e to indicate that there i s a very close relationship between the Kodaira dimension o f a threefold ( a property o f a threefold fro m classification theory) and the existence o f rational curves. Thes e problems are very interestin g but they also seem quite hard. Ou r contribution i n this directio n is mostly limite d to presenting som e examples and conjectures. In the second direction, one of the questions we were interested i n was that of understanding rationa l curve s on quintic hypersurfaces i n P4. Late r this was scaled down to understand lines o n quintic hypersurfaces i n P4, but even this seem s a hard problem. W e began to understand i t more completely onl y after the seminar had ended (se e [J ] ) . Very littl e time was lef t to consider the third direction. W e were fortunat e to have a series of lectures, but we could not pursue this interestin g an d important directio n i n any detail. The styl e of the seminar s was very informal . W e tried to keep them discussion-and-problem oriented . Note s were taken by H. Clemens who typed them up by the next day. Thes e notes constituted th e firs t version o f the present text. Durin g the seminar and afterwards, these notes were considerably revised , cut, expanded an d edited. Durin g this process we tried to keep the original informalit y o f the talks alive. The regular participants o f the semina r were J . Jimenez, T. Luo, K. Matsuki and the three o f us. Severa l other people joine d us fo r various lengt h of time. A hopefully complet e lis t is: 1 HIGHER DIMENSIONAL COMPLEX GEOMETRY J. Carlson, L. Ein, M. He, Y. Ma, D. Ortland, S . Pantazis, P. Roberts, D. Toledo, S. Turner, and Stephen Yau. W e are very- grateful fo r their contributio n t o the success of the seminar. We are especially thankful to those people who gave talks. The following i s a list of the lecture s of a mathematician othe r than one of the three of us. J. Carlson: Maxima l variations o f Hodge structures; L. Ein: Submanifold s o f generic complet e intersection s i n Grassmanians; L. Ein: A theorem of Gruson-Lazarsfeld-Peskine an d a lemma of Lazarsfeld; K. Matsuki: Con e Theorem ; K. Matsuki: Non-vanishin g Theorem ; D. Toledo: Kahle r structure s o n locally symmetri c spaces; D. Toledo: Proo f o f Sampson's theorem ; D. Toledo: Abelia n subalgebra s o f Lie algebras. At the final editing o f these notes som e talks were lef t out. This was the fat e of the followin g talks: H. Clemens: Abel-Jacob i maps; S. Turner: Ellipti c surface s i n characteristic p ; S. Yau: Eule r characteristic o f Chow varieties. These talks were about topics that we had no time to pursue further, and therefore they did not fi t neatly int o the fina l version of the notes. Our aim was to keep the notes advanced enoug h to be of interest eve n to the specialists, but understandable enoug h s o that a person with a good general background i n algebraic geometr y would be able to understand an d enjoy them. Especiall y a t the beginning, the lectures ar e rather informal and concentrate o n the geometric picture rathe r than o n a proof that i s correct i n every technical detail. W e hope that this informa l introductio n to [Ml ] will be helpful. Thes e matters occupy the firs t two lectures. The classification theor y o f surfaces i s reviewed fro m the point o f view of threefold theory i n Lecture 3 . Thi s lead s naturally to the next lectur e which i s an introduction t o the study of cones of curves. Lectur e 5 discusses the aims of Mori's program i n more detail, concentrating mainly o n flips, the presence o f which i s perhaps the most importan t differenc e between algebraic geometry i n two and i n three dimensions. A t the end of this lecture, a table compares the basic results i n the birational geometry o f surfaces and threefolds. Eve n though the lis t was selected with bias, the similaritie s ar e striking. Lecture 6 is a little more technical. I t discusses the singularities tha t arise naturally i n the stud y of smooth threefolds. Thes e are the three dimensional analogues o f the rational double points of surfaces. Thei r structur e i s however more complicated an d not completely known. 2 INTRODUCTION Lecture 7 discusses extensions of the Cone Theorem to relative situations an d equivariant settings . I n Lecture 8 we give quick proofs o f some vanishing theorems that are needed fo r the proof of the Cone Theorem. This lead s directly t o the next big section, which i s the proof o f the general Cone Theorem. Thi s i s done i n Lectures 9-13 . Here the proofs are (o r at least are intented to be) also technically correct . Th e proof of the final step (give n in Lecture 11 ) i s new, and makes i t possible t o avoid the rathe r technical relativ e case . A t least fo r us, this made the proof much clearer. The end of the firs t part o f these notes i s a discussion o f flops and flips. I f a rational curve on a quintic threefold i n P4 can be contracted, then i t can be flopped. Thu s understandin g flops yields results about rational curve s o n quintic threefold s in P4 . Th e simples t questio n to which this approach lead s is: Is i t true that, i f C is a smooth rational curve on a quintic threefold i n P4 which has normal bundle 0t(2) +0C(-4), then some multiple o f C moves? There ar e no such lines, but this situation ca n occur fo r plane conies o n some special quintic threefolds i n P4. ( A negative answer to the question i n that cas e appears i n [C3] , written afte r the completion o f these notes.) Two lectures are devoted to flips. Lectur e 1 4 is a general introduction, and Lecture 1 5 is an essentially complet e proof of the local description o f a threefold alon g a contractible rationa l curve that has negative intersectio n wit h the canonical class. This shoul d give a fairly clear ide a of the content o f the firs t seven chapter s of [M3] , and should enable the reader to go directly to Chapter 8 (afte r reviewing som e additional definition s and statements).
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