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We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective klt pair (X,D) with Kx + D big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of D and the volume of . We further show that all n-dimensional projective klt pairs (X,D), such that Kx + D is big and nef of fixed volume and such that the coefficients of D are contained in a given DCC set, form a bounded family. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family.
|Number of pages||27|
|Journal||Annales Scientifiques de l'École Normale Supérieure|
|Publication status||Accepted/In press - 11 Oct 2018|