Around and beyond the canonical class

This survey is an invitation to recent developments in higher dimensional birational geometry.Comment: to appear in "Birational geometry, rational curves, and arithmetic, Simons Symposium Proceedings", Springer Verla

On the Cone conjecture for Calabi-Yau manifolds with Picard number two

Following a recent work of Oguiso, we calculate explicitly the groups of automorphisms and birational automorphisms on a Calabi-Yau manifold with Picard number two. When the group of birational automorphisms is infinite, we prove that the Cone conjecture of Morrison and Kawamata holds.Comment: to appear in Math. Res. Let

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

New outlook on the minimal model program, I

We give a new and self-contained proof of the finite generation of adjoint rings with big boundaries. As a consequence, we show that the canonical ring of a smooth projective variety is finitely generated.Comment: to appear in Duke Math.

The Minimal Model Program Revisited

We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.Comment: to appear in Contributions to Algebraic Geometry, EMS Series of Congress Report

Abundance for varieties with many differential forms

We prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. This implies, for instance, that under this condition, hermitian semipositive canonical divisors are almost always semiample, and that klt pairs whose underlying variety is uniruled have good models in many circumstances. When the numerical dimension of $K_X$ is $1$, our results hold unconditionally in every dimension. We also treat a related problem on the semiampleness of nef line bundles on Calabi-Yau varieties