Related papers: A Matsushima theorem for K-polystable polarised sm…

We prove that all smooth Fano threefolds with Picard rank 2 and degree 28 are K-polystable, except for some explicit cases which we describe. We also give a classification of the normal bundle of a rational normal quartic curve in a smooth…

We prove that, for a spherical Fano threefold not in the Mori-Mukai family 2-29, and a weight function associated with the action of the connected center of a Levi subgroup of its automorphism group, weighted K-polystability is equivalent…

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the…

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…

We establish two consequences of the Kawamata--Morrison--Totaro cone conjecture, and prove them unconditionally in all dimensions. First, for a K-trivial variety, the natural action of its automorphism group on the set of ample divisor…

For the Calabi-Yau threefolds $X$ constructed by C. Schoen as fiber products of generic rational elliptic surfaces, we show that the action of the automorphism group of $X$ on the K\"ahler cone of $X$ has a rationally polyhedral fundamental…

Let $X$ be a smooth Fano threefold. We show that $X$ admits a non-isomorphic surjective endomorphism if and only if $X$ is either a toric variety or a product of $\mathbb{P}^1$ and a del Pezzo surface; in this case, $X$ is a rational…

Let $(X,L)$ be any Fano manifold polarized by a positive multiple of its fundamental divisor $H$. The polynomial defining the Hilbert curve of $(X,L)$ boils down to being the Hilbert polynomial of $(X,H)$, hence it is totally reducible over…

We prove that every smooth Fano threefold from the family No 2.8 is K-stable. Such a Fano threefold is a double cover of the blow-up of $\mathbb{P}^3$ at one point branched along an anti-canonical divisor.

We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on Q-Fano threefolds, Gorenstein log del Pezzo surfaces and P^1. Similar results are obtained…

We prove that there exists a positive number $C_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over {\bf C}, the automorphism group $Aut(X)$ satisfies the inequality $\sharp{Aut}(X)\leq…

Consider a smooth connected algebraic group $G$ acting on a normal projective variety $X$ with an open dense orbit. We show that Aut($X$) is a linear algebraic group if so is $G$; for an arbitrary $G$, the group of components of Aut($X$) is…

We present an elementary way of recovering a well-known criterion of K-stability for Fano reductive group compactifications.

We construct proper moduli algebraic spaces of K-polystable $\mathbb{Q}$-Fano cones (a.k.a. Calabi-Yau cones) or equivalently their links i.e., Sasaki-Einstein manifolds with singularities. As a byproduct, it gives alternative algebraic…

Let $X$ be a (smooth) compact complex surface. We show that the torsion subgroup of the biholomorphic automorphism group $\operatorname{Aut}(X)$ is virtually nilpotent. Moreover, we study the Tits alternative of $\operatorname{Aut}(X)$ and…

We show that if a Fano manifold has discrete automorphism group and admits a polarized K\"ahler-Einstein metric, then there exists a sequence of anticanonically balanced metrics converging smoothly to the K\"ahler-Einstein metric. Our proof…

It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this…

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford…

For a given polarized toric variety, we define the notion of $\lambda$-stability which is a natural generalization of uniform K-stability. At the neighbourhoods of the vertices of the corresponding moment polytope $\Delta$, we consider…

We introduce a new subclass of Fano varieties (Casagrande-Druel varieties), that are $n$-dimensional varieties constructed from Fano double covers of dimension $n-1$. We conjecture that a Casagrande-Druel variety is K-polystable if the…