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

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair $(X, \Delta)$ have finite, rational…

The wonderful compactification $X_m$ of a symmetric homogeneous space of type AIII$(2,m)$ for each $m \geq 4$ is Fano, and its blowup $Y_m$ along the unique closed orbit is Fano if $m \geq 5$ and Calabi-Yau if $m = 4$. Using a combinatorial…

The main goal of this paper is to prove the polystability of the logarithmic tangent sheaf $\mathscr T_X(-D)$ of a log canonical pair $(X,D)$ whose canonical bundle $K_X+D$ is ample, generalizing in a significant way a theorem of Enoki. We…

Let $(X, \Delta)$ be a log Fano pair with standard coefficients endowed with a singular K\"ahler--Einstein metric. We show that the adapted tangent sheaf $\mathcal{T}_{X, \Delta, f}$ and the adapted canonical extension $\mathcal{E}_{X,…

We introduce a notion of K-stability for adjoint foliated structures via test configurations and the foliated Donaldson-Futaki invariant. We prove reduction to special test configurations for adjoint Fano foliated structures by showing that…

In this paper, with the aim of establishing a structure theorem for a compact K\"ahler manifold $X$ with semi-positive holomorphic sectional curvature, we study a morphism $\phi: X \to Y$ to a compact K\"ahler manifold $Y$ with…

We prove that the birational automorphism group of a general Calabi-yau complete intersection $X$ given by ample divisors in $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_l}$ is always Lorentzain. Applying the Kawamata-Morrison cone…

Let $X$ be a compact normal K\"ahler space whose canonical sheaf is a rank-one free $\mathcal O_X$ module and whose singularities are isolated, rational and quasi-homogeneous. We prove then that under a topological hypothesis the…

In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk…

Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata…

In this paper, we shall show that a polarized algebraic manifold is K-stable if the polarization class admits a Kaehler metric of constant scalar curvature. This generalizes the results of Chen-Tian, Donaldson and Stoppa. (Parts of the…

We prove that the automorphism group $\mathrm{Aut}(X)$ of an affine spherical variety $X$ acts transitively on the set of smooth points $X^{reg}.$ If every invertible regular function on $X$ is constant, we prove that $X$ is flexible, i.e.,…

We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider a particular family of anti-solvable finite groups containing the simple alternating groups for $n\neq 6$ and all 26 sporadic simple groups.…

A K3 surface $X$ over a $p$-adic field $K$ is said to have good reduction if it admits a proper smooth model over the ring of integers of $K$. Assuming this, we say that a subgroup $G$ of $\mathrm{Aut}(X)$ is extendable if $X$ admits a…

In this paper we prove that over algebraically closed field $K$ of positive characteristic $\neq 2$ every automorphism of the group of origin-preserving automorphisms of the polynomial algebra $K[x_1,\ldots, x_n]$ ($n>3$) which fixes every…

Let M be a (possibly non-orientable) compact 3-manifold with (possibly empty) boundary consisting of tori and Klein bottles. Let $X\subset\partial M$ be a trivalent graph such that $\partial M\setminus X$ is a union of one disc for each…

Olsson showed in [Ols25] that if $\mathcal{X} \to X$ is a $\mathbf{G}_m$-gerbe over a smooth projective variety over an algebraically closed field $k$ such that the Brauer class of $\mathcal{X}$ has order prime to the characteristic of $k$,…

In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural…

Tian's criterion for K-stability states that a Fano variety of dimension $n$ whose alpha invariant is greater than $\frac{n}{n+1}$ is K-stable. We show that this criterion is sharp by constructing singular Fano varieties with alpha…

Let $X:=\mathbb{A}^{n}_{R}$ be the $n$-dimensional affine space over a discrete valuation ring $R$ with fraction field $K$. We prove that any pointed torsor $Y$ over $\mathbb{A}^{n}_{K}$ under the action of an affine finite type group…