Vladimir Lazić

New outlook on the Minimal Model Program, II

We prove that the finite generation of adjoint rings implies all the foundational results of the Minimal Model Program: the Rationality, Cone and Contraction theorems, the existence of flips, and termination of flips with scaling in the presence of a big boundary.

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.

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.

The Minimal Model Program revisited

This survey paper is based on my IMPANGA lectures given in the Banach Center, Warsaw in January 2011. We study the moduli of holomorphic map germs from the complex line into complex compact manifolds with applications in global singularity theory and the theory of hyperbolic algebraic varieties.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.We introduce a variety

Around and Beyond the Canonical Class

\hat{G}_2

Finite generation and geography of models

parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety

Adjoint rings are finitely generated

G_2

New outlook on Mori theory, I

, the adjoint variety of the simple Lie group

Abundance for varieties with many differential forms

\mathbb{G}_2

Automorphisms of Calabi-Yau threefolds with Picard number three

. It occurs that it is also the image of