A note on minimal models for pmp actions

Given a countable group $G$, we say that a metrizable flow $Y$ is model-universal if by considering the various invariant measures on $Y$, we can recover every free measure-preserving $G$-system up to isomorphism. Weiss has constructed a minimal model-universal flow. In this note, we provide a new, streamlined construction, allowing us to show that a minimal model-universal flow is far from unique

The structure of algebraic varieties

ICM lecture on minimal models and moduli of varieties

On Minimal Models and the Termination of Flips for Generalized Pairs

Das Ziel dieser Arbeit ist die Untersuchung zweier offener Probleme in der höherdimensionalen birationalen Geometrie, nämlich der Vermutung zur Existenz von minimalen Modellen und der Vermutung zur Terminierung von Flips. Wir arbeiten haupts ächlich mit verallgemeinerten Paaren und untersuchen demzufolge die entsprechenden Versionen der oben genannten Vermutungen des Minimal-Modell-Programms in diesem breiteren Rahmen. Der erste Teil der Dissertation widmet sich daher der Entwicklung der grundlegenden Aspekte der Theorie der verallgemeinerten Paare. Um die Vermutung zur Existenz von minimalen Modellen anzugehen, betrachten wir zunächst bestimmte Zariski-Zerlegungen in höheren Dimensionen, die sogennanten schwachen Zariski-Zerlegungen und NQC Nakayama-Zariski-Zerlegungen. Anschließend beweisen wir, dass die Existenz von minimalen Modellen für log-kanonische verallgemeinerte Paare aus der Existenz von minimalen Modellen für glatte Varietäten folgt, und ferner, dass die Existenz von minimalen Modellen im Wesentlichen zur Existenz dieser Zariski-Zerlegungen äquivalent ist. Der letzte Teil dieser Arbeit befasst sich mit der Vermutung zur Terminierung von Flips. Wir zeigen zuerst die Spezielle Terminierung für log-kanonische verallgemeinerte Paare. Danach beweisen wir die Terminierung von Flips für log-kanonische verallgemeinerte Paare der Dimension 3 sowie für pseudo-effektive log-kanonische verallgemeinerte Paare der Dimension 4.The aim of this thesis is the investigation of two open problems in higher-dimensional birational geometry, namely the existence of minimal models conjecture and the termination of ips conjecture. We mainly work with generalized pairs and we therefore study the corresponding versions of the aforementioned conjectures of the Minimal Model Program in this wider context. Consequently, the first part of the thesis is devoted to the development of the basic aspects of the theory of generalized pairs. In order to deal with the existence of minimal models conjecture, we first study particular Zariski decompositions in higher dimensions, the so-called weak Zariski decompositions and NQC Nakayama-Zariski decompositions. Subsequently, we prove that the existence of minimal models for log canonical generalized pairs follows from the existence of minimal models for smooth varieties, and we also demonstrate that the existence of minimal models is essentially equivalent to the existence of those Zariski decompositions. The last part of this thesis focuses on the termination of ips conjecture. First, we show the special termination for log canonical generalized pairs. Afterwards, we establish the termination of ips for log canonical generalized pairs of dimension 3 as well as for pseudo-effective log canonical generalized pairs of dimension 4

On the cohomology algebra of a fiber

Let f:E-->B be a fibration of fiber F. Eilenberg and Moore have proved that there is a natural isomorphism of vector spaces between H^*(F;F_p) and Tor^{C^*(B)}(C^*(E),F_p). Generalizing the rational case proved by Sullivan, Anick [Hopf algebras up to homotopy, J. Amer. Math. Soc. 2 (1989) 417--453] proved that if X is a finite r-connected CW-complex of dimension < rp+1 then the algebra of singular cochains C^*(X;F_p) can be replaced by a commutative differential graded algebra A(X) with the same cohomology. Therefore if we suppose that f:E-->B is an inclusion of finite r-connected CW-complexes of dimension < rp+1, we obtain an isomorphism of vector spaces between the algebra H^*(F;F_p) and Tor^{A(B)}(A(E),F_p) which has also a natural structure of algebra. Extending the rational case proved by Grivel-Thomas-Halperin [PP Grivel, Formes differentielles et suites spectrales, Ann. Inst. Fourier 29 (1979) 17--37] and [S Halperin, Lectures on minimal models, Soc. Math. France 9-10 (1983)] we prove that this isomorphism is in fact an isomorphism of algebras. In particular, $H^*(F;F_p) is a divided powers algebra and p-th powers vanish in the reduced cohomology \mathaccent "707E {H}^*(F;F_p).Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-36.abs.htm

On Minimal Models and Canonical Models of Elliptic Fourfolds with Section

One of the main research programs in Algebraic Geometry is the classification of varieties. Towards this goal two methodologies arose, the first is classifying varieties up to isomorphism which leads to the study of moduli spaces and the second is classifying varieties up to birational equivalences which leads to the study of birational geometry. Part of the engine of the birational classification is the Minimal Model Program which, given a variety, seeks to find "nice" birational models, which we call minimal models. Towards this direction much progress has been made but there is also much to be done. One aspect of interests is the role of algebraic fiber spaces as the end results of the Minimal Model Program are categorized into Mori fiber spaces, Iitaka fibrations over canonical models and varieties of general type. A natural problem to consider is, starting with an algebraic fiber space, how might it behave with respect to the Minimal Model Program. For case of elliptic threefolds, it was shown by Grassi, that minimal models of elliptic threefolds relate to log minimal models of the base surface. This shows that minimal models, in a sense, have to respect the fiber structure for elliptic threefolds. In this dissertation, I will provide a framework towards a generalization for higher dimensional elliptic fibration and along the way recover the results of Grassi for elliptic fourfolds with section

On the existence of minimal models for log canonical pairs

We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.Comment: v3: minor changes; title changed; to appear in Publ. Res. Inst. Math. Sc