Related papers: On a connectedness principle of Shokurov-Koll\'ar …

Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$. A conjecture, known as the Shokurov-Koll\'{a}r connectedness principle, predicts that $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at…

A conjecture, known as the Shokurov-Koll\'ar connectedness principle, predicts the following. Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s \in S$, the…

We study the non-klt locus of singularities of pairs. We show that given a pair $(X,B)$ and a projective morphism $X\to Z$ with connected fibres such that $-(K_X+B)$ is nef over $Z$, the non-klt locus of $(X,B)$ has at most two connected…

Following Shokurov's ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space. Thus, in…

It is well known that a smooth projective Fano variety is rationally connected. Recently Zhang (and later Hacon and McKernan as a special case of their work on the Shokurov RC-conjecture) proved that the same conclusion holds for a klt pair…

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(K+D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and…

We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a…

Let $X\rightarrow S$ be a fibration of relative dimension at most two and let $(X,\Delta)$ be a klt pair for which $K_X+\Delta \equiv_S 0$. We show that there are only finitely many Mori chambers and Mori faces in the movable effective cone…

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.

Let $(X,B)$ be an $\epsilon$-lc pair of dimension $d$ with a closed point $x\in X$. Birkar and Shokurov conjectured that there is an effective Cartier divisor $H$ passing through $x$ such that $(X,B+tH)$ is lc near $x$, where $t$ is a…

Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. First, we show the ampleness of $K_X+tL_X$ with sufficiently large $t$ if either the Kodaira dimension $\kappa(X)\neq 0$…

We prove the termination of 4-fold log flips for klt pairs of Kodaira dimension $\kappa\ge 2$.

Recently Swatee Naik and Theodore Stanford proved that two S-equivalent knots are related by a finite sequence of doubled-delta moves on their knot diagrams. We show that classical S-equivalence is not sufficient to extend their result to…

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.…

Let $(X,\Delta)$ be a smooth complex projective simple normal crossing pair of dimension $n\geq 3$ endowed with an everywhere nondegenerate logarithmic conformal tensor. If $K_X+\Delta$ is not nef, then precisely one of the following…

In this paper, by running MMP and considering the anti-canonical fibration, we prove the Morrison-Kawamata cone conjecture for klt Calabi-Yau pairs $(X,\Delta)$ such that $\dim X$ is at most $4$, and the Iitaka dimension $\kappa(X,-K_X)$ is…

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…

Let $(X,\Delta)$ be a log canonical $4$-fold over an algebraically closed field of characteristic zero. We prove that any sequence of $(K_X+\Delta)$-flips terminates.

We prove a conjecture of Koll\'ar stating that the local fundamental group of a klt singularity $x$ is finite. In fact, we prove a stronger statement, namely that the fundamental group of the smooth locus of a neighbourhood of $x$ is…

A conjecture of Luo, Tian and Wu (2022) says that for every positive integer $k$ and every finite tree $T$ with bipartition $X$ and $Y$ (denote $t = \max\{|X|,|Y |\})$, every $k$-connected bipartite graph $G$ with $\delta(G) \geq k + t$…