Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.

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Existence of minimal models for varieties of log general type

In this paper, we prove the cone theorem and the contraction theorem for pairs (X;B), where X is a normal variety and B is an effective R-divisor on X such that KX +B is R-Cartier.

/pdf/fundamental-theorems-for-the-log-minimal-model-program-1p9s7lsyi3.pdf

Fundamental Theorems for the Log Minimal Model Program

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 positive integer n there exists an integer r
 n
 such that if X is a smooth projective variety of general type and dimension n, then $\phi_{rK_X}\colon X\dasharrow\mathbb{P}(H^0(\mathcal{O}_{X}(rK_X)))$
 is birational for all r≥r
 n
 .

/pdf/boundedness-of-pluricanonical-maps-of-varieties-of-general-1xjxbmc2so.pdf

Boundedness of pluricanonical maps of varieties of general type

We prove the rational connectedness conjecture of V. V. Shokurov in [20] which, in particular, implies that the fibres of a resolution of a variety with divisorial log terminal singularities are rationally chain connected

/pdf/on-shokurov-s-rational-connectedness-conjecture-2o2a6u9zqs.pdf

On Shokurov's rational connectedness conjecture

A recurring difficulty in the Minimal Model Program (MMP) is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more complicated; they need not even be CohenMacaulay. The aim of this paper is to prove that, as conjectured in [Kol92, 1.13], log canonical singularities are Du Bois. The concept of Du Bois singularities, abbreviated as DB, was introduced by Steenbrink in [Ste83] as a weakening of rationality. It is not clear how to define Du Bois singularities in positive characteristic, so we work over a field of characteristic 0 throughout the paper. The precise definition is rather involved, see (1.10), but our main applications rely only on the following consequence:

/pdf/log-canonical-singularities-are-du-bois-1y1xk19cnb.pdf

Log canonical singularities are Du Bois

We extend the Cone Theorem of the Log Minimal Model Program to log varieties with arbitrary singularities

Quasi-log varieties

For a birational analogue of minimal elliptic surfaces X/Y, the singularities of the fibers define a log structure in codimension one on Y. Via base change, we have a log structure in codimension one on Y', for any birational model Y' of Y. We show that these codimension one log structures glue to a unique log structure, defined on some birational model of Y (Shokurov's BP Conjecture). We have three applications: inverse of adjunction for the above mentioned fiber spaces, the invariance of Shokurov's FGA-algebras under restriction to exceptional lc centers, and a remark on the moduli part of parabolic fiber spaces.

/pdf/shokurov-s-boundary-property-48n0fcen60.pdf

Shokurov's Boundary Property

We prove the existence of ladders on log Fano varieties of coindex less than 4, as an application of adjunction and nonvanishing.

Ladders on Fano varieties

We discuss adjunction formulas for fiber spaces and embeddings, extending the known results along the lines of the Adjunction Conjecture, independently proposed by Y. Kawamata and V.V. Shokurov. 
As an application, we simplify Koll\'ar's proof for the Anghern and Siu's quadratic bound in the Fujita's Conjecture. We also connect adjunction and its precise inverse to the problem of building isolated log canonical singularities.

The Adjunction Conjecture and its applications

The LCS locus is an essential ingredient in the proof of fundamental results of Log Minimal Model Program, such as nonvanishing and base point freeness theorems. We prove in this paper that the LCS locus of a log canonical variety has seminormal singularities.

Florin Ambro

Papers

Quasi-log varieties

Shokurov's Boundary Property

Ladders on Fano varieties

The Adjunction Conjecture and its applications

The locus of log canonical singularities