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Journal ArticleDOI

New directions in the Minimal Model Program

01 Mar 2021-Bollettino Della Unione Matematica Italiana (Springer Science and Business Media LLC)-Vol. 14, Iss: 1, pp 179-190
TL;DR: In this article, a survey of recent developments in the Minimal Model Program is presented, focusing on its generalisations to the category of foliated varieties and the categories of varieties defined over any algebraically closed field of positive characteristic.
Abstract: We survey some recents developments in the Minimal Model Program. After an elementary introduction to the program, we focus on its generalisations to the category of foliated varieties and the category of varieties defined over any algebraically closed field of positive characteristic.

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Journal ArticleDOI
TL;DR: In this paper, the Ricci form of some Kahler metric is shown to be closed and its cohomology class must represent the first Chern class of M. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.
Abstract: Therefore a necessary condition for a (1,l) form ( G I a ' r r ) I,,, Rlr dz' A d? to be the Ricci form of some Kahler metric is that it must be closed and its cohomology class must represent the first Chern class of M. More than twenty years ago, E. Calabi [3] conjectured that the above necessary condition is in fact sufficient. This conjecture of Calabi can be reduced to a problem in non-linear partial differential equation.

2,903 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that pl-flips exist in dimension n − 1, assuming finite generation in dimension N − 1 and assuming that pl flips exist in all dimensions.
Abstract: Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

1,612 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense, which disproves a long-standing conjecture that a compact KG admits KG metrics if it has positive first Chern class and no nontrivial holomorphic vector fields.
Abstract: In this paper, we prove that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense. In particular, this disproves a long-standing conjecture that a compact Kahler manifold admits Kahler-Einstein metrics if it has positive first Chern class and no nontrivial holomorphic vector fields. We will also establish an analytic criterion for the existence of Kahler-Einstein metrics. Our arguments also yield that the analytic criterion is satisfied on stable Kahler manifolds, provided that the partial C 0-estimate posed in [T6] is true.

1,038 citations

Journal ArticleDOI
TL;DR: In this article, it was shown that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings.
Abstract: We prove that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H. Luo. This gives, as a Corollary, the uniqueness of constant scalar curvature Kahler metrics in a given rational cohomology class. The proof uses results in the literature on the asymptotics of the Bergman kernel. The arguments are presented in a general framework involving moment maps for two different group actions.

734 citations

Journal ArticleDOI
TL;DR: In this paper, the authors studied the study of surface singularities using recent advances in 3D geometry and proved the existence of a minimal resolution of singularities for a given set of surfaces.
Abstract: 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: t e A } . Then the total space X = U X 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. [-Ko3, KMM]). After substantial contributions by several mathematicians (Benveniste, Kawamata, Kollfir, Mori, Reid, Shokurov, Vichweg) this was recently completed by Mori [Mo 3]. A special case, which is nonetheless sufficient for the applications presented here, was settled by several persons. A proof was first announced by Tsunoda [TsM], later followed by Shokurov [Sh], Mori [Mo2] and Kawamata [Kaw2]. 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: s e S } is not a flat family of surfaces, and it is of interest to find necessary and sufficient conditions for this to hold.

619 citations