Related papers: The minimal model program for b-log canonical divi…

We survey our recent papers (some being joint ones) about the relation between the geometry of a compact K\"ahler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program…

Minimal model conjecture for a proper variety $X$ is that if $\kappa(X)\geq 0$, then $X$ has a minimal model with the abundance and if $\kappa =-\infty$, then $X$ is birationally equivalent to a variety $Y$ which has a fibration $Y \to Z$…

We examine the noncommutative minimal model program for orders on arithmetic surfaces, or equivalently, arithmetic surfaces enriched by a Brauer class $\beta$. When $\beta$ has prime index $p>5$, we show the classical theory extends with…

We discuss the relationship among various conjectures in the minimal model theory including the finite generation conjecture of the log canonical rings and the abundance conjecture. In particular, we show that the finite generation…

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of…

R. B. Melrose's b-calculus provides a framework for dealing with problems of partial differential equations that arise in singular or degenerate geometric situations. This article is a somewhat informal short course introducing many of the…

In this article we define generalized pairs $(X, B+\boldsymbol{\beta})$ where $X$ is an analytic variety and $\boldsymbol{\beta}$ is a b-(1,1) current. We then prove that almost all standard results of the MMP hold in this generality for…

Given an arbitrary complex-valued infinite matrix A and a positive integer n we introduce a naturally associated polynomial basis B_A of C[x0...xn]. We discuss some properties of the locus of common zeros of all polynomials in B_A having a…

We prove that the canonical ring of a smooth projective variety is finitely generated.

For moduli of polarized smooth K-trivial a.k.a., Calabi-Yau varieties in a general sense, we revisit a classical problem of constructing its "weak K-moduli" compactifications which parametrizes K-semistable (i.e., semi-log-canonical…

We show the semi-continuity property of minimal log discrepancies for varieties which have a crepant resolution in the category of Deligne-Mumford stacks. Using this property, we also prove the ideal-adic semi-continuity problem for toric…

We show that given any two minimal models of a generalized lc pair, there exist small birational models which are connected by a sequence of symmetric flops. We also present some applications.

We prove the existence of pl-flips.

We propose a subconjecture that implies the semiampleness conjecture for quasi-numerically positive log canonical divisors and prove the semiampleness in some elementary cases.

These notes survey some basic results in toric varieties over a field with examples and applications. A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any…

We prove that many of the results of the LMMP hold for $3$-folds over fields of characteristic $p>5$ which are not necessarily perfect. In particular, the existence of flips, the cone theorem, the contraction theorem for birational extremal…

This semi-expository paper discusses the log minimal model program as applied to the moduli space of curves, especially in the case of curves of genus two. Log canonical models for these moduli spaces can often be constructed using the…

In this article we show that the Log Minimal Model Program holds for $\mathbb{Q}$-factorial lc pair $(X,\Delta)$ with $X$ being a compact K\"ahler $3$-fold having only klt singularities.

We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli are also considered.

We prove a structure theorem for the Albanese maps of varieties with Q-linearly trivial log canonical divisors. Our start point is the action of a nonlinear algebraic group on a projective variety.