On the number and boundedness of log minimal models of general type

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, Δ) with KX+ Δ big, we show more generally that the number of its weak log canonical models can be bounded in terms of the coefficients of Δ and the volume of KX+ Δ. We further show that all n-dimensional projective klt pairs (X, Δ), such that KX+ Δ is big and nef of fixed volume and such that the coefficients of Δ are contained in a given DCC set, forma bounded family. It follows that in any dimension, minimal models of general type and bounded volume form a bounded family.