János Kollár

Rationally Connected Varieties

This is the key chapter of the book. Its aim is to study the birational properties of those varieties which are covered by rational curves. It is reasonable to expect that many of their properties can be detected by studying the rational curves on them.

Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds

Abstract 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 Kahler–Einstein metrics on certain Fano orbifolds, following Nadels original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, three new examples of rigid Kahler–Einstein Del Pezzo surfaces with quotient singularities are obtained.

Sharp effective Nullstellensatz

The usual proofs of this result, however, give no information about the g1s; for instance they give no bound on their degrees. This question was first considered by G. Hermann [H]. She used elimination theory to get a bound on the degree of the gis which was doubly exponential in the number of variables. Her results were later improved in [MW] and in [Th]. All these produce bounds that are doubly exponential in the number of variables. A major breakthrough was achieved by Brownawell [B1] who proved the following result:

Singularities of the minimal model program

Preface Introduction 1. Preliminaries 2. Canonical and log canonical singularities 3. Examples 4. Adjunction and residues 5. Semi-log-canonical pairs 6. Du Bois property 7. Log centers and depth 8. Survey of further results and applications 9. Finite equivalence relations 10. Appendices References Index.

Log canonical singularities are Du Bois

A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be Cohen-Macaulay. The aim of this paper is to prove that log canonical singularities are Du Bois. The concept of Du Bois singularities, introduced by Steenbrink, is a weakening of rationality. We also prove flatness of the cohomology sheaves of the relative dualizing complex of a projective family with Du Bois fibers. This implies that each connected component of the moduli space of stable log varieties parametrizes either only Cohen-Macaulay or only non-Cohen-Macaulay objects.

Norm-graphs and bipartite turán numbers

For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|≥ct·n2−1/t which do not contain a complete bipartite graghKt,t!+1 This establishes the exact order of magnitude of the Turán numbers ex (n, Kt,s) for any fixedt and alls≥t!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.

QUOTIENT SPACES MODULO ALGEBRAIC GROUPS

The paper proves that if a reductive group scheme acts properly on a scheme then the geometric quotient exists as an algebraic space. As a consequence we obtain the existence of the moduli spcace of canonically polarized varieties over Spec Z.

The Nash problem on arc families of singularities

Nash [21] proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution. He asked if the converse also holds: Does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.

The structure of algebraic threefolds: an introduction to Mori's program

Introduction. This article intends to present an elementary introduction to the emerging structure theory of higher-dimensional algebraic varieties. Introduction is probably not the right word; it is rather like a travel brochure describing the beauties of a long cruise, but neglecting to mention that the first half of the trip must be spent toiling in the stokehold. Perusal of brochures might give some compensation for lack of royal roads. Having this limited aim in mind, the prerequisities were kept very low. As a general rule, geometry is emphasized over algebra. Thus, for instance, nothing is used from abstract algebra. This had to be compensated by using more results from topology and complex variables than is customary in introductory algebraic geometry texts. Still, aside from some harder results used in occasional examples, only basic notions and theorems are required. Throughout the history of algebraic geometry the emphasis constantly shifted between the algebraic and the geometric sides. The first major step was a detailed study of algebraic curves by Riemann. He approached the subject from geometry and analysis, and gave a quite satisfactory structure theory. Subsequently the German school, headed by Max Noether, introduced algebra to the subject and problems arising from algebraic geometry substantially influenced the development of commutative algebra, especially the works of Emmy Noether and Krull. During the same period the Italian school of Castelnuovo, Enriques, and Severi investigated the geometry of algebraic surfaces and achieved a satisfactory structure theory. Their work, however, lacked the Hilbertian rigor, and

Continuous rational functions on real and \(p\)-adic varieties

Strictly speaking, a rational function f is not really a function on R in general since it is defined only on the dense open set where q 6= 0. Nonetheless, even if q vanishes at some points of R, it can happen that there is an everywhere defined continuous function f that agrees with f at all points where f is defined. Such an f is unique. For this reason, we identify f with f and call f itself a continuous rational function on R. For instance,