Introduction and statement of results Glossary of notation and conventions Gorenstein del Pezzo surfaces Bug-eyed covers Log deformation theory Criteria for log uniruledness Reduction to… Expand

Using the techniques of [20] and [10], we prove that certain log forms may be lifted from a divisor to the ambient variety. As a consequence of this result, following [22], we show that: For any… Expand

We prove a conjecture of V. V. Shokurov which in particular implies that the fibers of a resolution of a variety with divisorial log terminal singularities are rationally chain connected.

We show that the number of birational automorphisms of a variety of general type X is bounded by c vol(X;KX), where c is a constant that only depends on the dimension of X.

Abstract.We prove that the set of accumulation points of thresholds in dimension three is equal to the set of thresholds in dimension two, excluding one.

We prove a conjecture of Batryev which states that the family of all Fano varieties with kawamata log terminal singularities and fixed index, forms a bounded family.