Related papers: Anticanonical minimal models and Zariski decomposi…

We introduce linearly decomposable (LD) generalized pairs, which serve as a workable substitute for rational decompositions in the non-NQC setting. Using LD generalized pairs, together with a refinement of special termination and…

We give a complete classification of reductive symmetric pairs (g, h) with the following property: there exists at least one infinite-dimensional irreducible (g,K)-module X that is discretely decomposable as an (h,H \cap K)-module. We…

Given a complex-projective klt pair $(X, \Delta)$ with standard coefficients and such that $K_X + \Delta$ is ample, we determine necessary and sufficient conditions for the pair $(X, \Delta)$ to be uniformized by a bounded symmetric domain.…

We compare the minimal model of a log canonical pair with the minimal model of its reduced boundary. These results are then used to study the existence of the minimal model of a semi-log-canonical pair using its normalization.

Using currents with minimal singularities, we construct pointwise minimal multiplicities for a real pseudo-effective $(1,1)$-class $\alpha$ on a compact complex $n$-fold $X$, which are the local obstructions to the numerical effectivity of…

We show that the existence of a birational weak Zariski decomposition for a pseudo-effective generalized polarized lc pair is equivalent to the existence of a generalized polarized log terminal model.

We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the…

Let $f:(X,B)\to Z$ be a 3-fold extremal dlt flipping contraction defined over an algebraically closed field of characteristic $p>5$, such that the coefficients of $\{B\}$ are in the standard set $\{1-\frac 1n|n\in \mathbb N\}$, then the…

Let $(X,\Delta)$ be a projective, log canonical, $K$-trivial pair over the complex numbers. Let $Z$ be a minimal log canonical center of $(X,\Delta)$ and suppose that there exists a torus $\mathbb{T}\subseteq\operatorname{Aut}(X)$…

Given an NQC log canonical generalized pair $(X,B+M)$ whose underlying variety $X$ is not necessarily $\mathbb{Q}$-factorial, we show that one may run a $(K_X+B+M)$-MMP with scaling of an ample divisor which terminates, provided that…

We study the minimal model program for lc pairs on projective morphism between complex analytic spaces. More precisely, we generalize the results by Birkar and the second author to the setup by Fujino.

In this paper, we show that for any projective klt pair $(X,\Delta)$ over an algebraically closed field of characteristic \(0\) and any big $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor $L$ on $X$, the invariants $\alpha(X,\Delta,L)$ and…

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 $\mathbb R$-divisor on $X$ such that $K_X+B$ is $\mathbb R$-Cartier.

We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$.…

Suppose that $X$ is a projective variety over an algebraically closed field of characteristic $p > 0$. Further suppose that $L$ is an ample (or more generally in some sense positive) divisor. We study a natural linear system in $|K_X + L|$.…

In this paper, we establish a structure theorem for projective klt pairs $(X,\Delta)$ with nef anti-log canonical divisor; specifically, we prove that, up to replacing $X$ with a finite quasi-\'etale cover, $X$ admits a locally trivial…

We prove $\mathbb{R}$-boundary divisor versions of results proved by Birkar or Hacon-Xu on special kinds of the relative log minimal model program.

We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$.…

In this paper, we find necessary and sufficient conditions to identify pairs of matrices $X$ and $Y$ for which there exists $\Delta \in \mathbb C^{n,n}$ such that $\Delta+\Delta^*$ is positive semidefinite and $\Delta X=Y$. Such a $\Delta$…

Let $(X/Z,B+A)$ be a $\Q$-factorial dlt pair where $B,A\ge 0$ are $\Q$-divisors and $K_X+B+A\sim_\Q 0/Z$. We prove that any LMMP$/Z$ on $K_X+B$ with scaling of an ample$/Z$ divisor terminates with a good log minimal model or a Mori fibre…