Related papers: Rational connectedness of log $Q$-Fano varieties
Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain a subvariety of codimension one…
This paper is concerned with a sufficient criterion to guarantee that a given foliation on a normal variety has algebraic and rationally connected leaves. Following ideas from a preprint of Bogomolov-McQuillan and using recent works of…
Varieties of Fano type are very well behaved with respect to MMP, and they are known to be rationally connected. We study the relation between classes of rationally connected varieties and varieties of Fano type. We give examples of…
Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote the support of the rounddown of D, and K the canonical divisor of X. We show that any smooth family of canonically polarized varieties over X\S is isotrivial if…
Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply…
We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.
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 the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.
We show that minimal models of log canonical pairs exist, assuming the existence of minimal models of smooth varieties.
We consider rationally connected complex projective manifolds M and show that their loop spaces--infinite dimensional complex manifolds--have properties similar to those of M. Furthermore, we give a finite dimensional application concerning…
We prove a structure theorem for non-isomorphic endomorphisms of weak Q-Fano threefolds, or more generally for threefolds with big anti-canonical divisor. Also provided is a criterion for a fibred rationally connected threefold to be…
In this short note, we give an alternative proof of the semipositivity of the Chow-Mumford line bundle for families of K-semistable log-Fano pairs, and of the nefness threeshold for the log-anti-canonical line bundle on families of K-stable…
We prove that the canonical ring of a smooth projective variety is finitely generated.
One of the central aims of the Minimal Model Program is to show that a projective log canonical pair $(X,\Delta)$ with $K_X+\Delta$ pseudoeffective has a good model, i.e.\ a minimal model $(Y,\Delta_Y)$ such that $K_Y+\Delta_Y$ is…
We show that any union of slc strata of a Fano log pair with semi-log canonical singularities is simply connected. In particular, Fano log pairs with semi-log canonical singularities are simply connected, which confirms a conjecture of the…
Motivated by the problem of classifying toric $2$-Fano manifolds, we introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension. This invariant $m(X)\in\{1, \dots,\dim(X)\}$ captures the minimal…
The notion of 'slope rational connectedness' is introduced in the context of smooth orbifold pairs. The main result parallels the characterization of the rational connectedness of projective manifolds in terms of either the non-existence of…
Let $f:X\to U$ be a projective morphism of normal varieties and $(X,\Delta)$ a dlt pair. We prove that if there is an open set $U^0\subset U$, such that $(X,\Delta)\times_U U^0$ has a good minimal model over $U^0$ and the images of all the…
We prove that a degeneration rationally connected varieties over a field of characteristic zero always contains a geometrically irreducible subvariety which is rationally connected.
We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a…