Related papers: Separatedness of moduli of K-stable varieties

We show that the CM line bundle on a proper family parametrizing specially K-stable varieties with maximal variation is ample. As an application, we show projectivity of any proper subspace of the coarse moduli space of uniformly…

The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of K\"ahler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting'…

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…

We express notions of K-stability of polarized spherical varieties in terms of combinatorial data, vastly generalizing the case of toric varieties. We then provide a combinatorial sufficient condition of G-uniform K-stability by studying…

We establish the geometric Shafarevich boundedness conjecture for the moduli stack of stable minimal models, including in particular the moduli stack of KSB pairs.

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…

In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which…

We prove that the degree of the CM line bundle for a normal family over a curve with fixed general fibers is strictly minimized if the special fiber is either a smooth projective manifold with a unique cscK metric or ``specially K-stable",…

We show boundedness of polarized Calabi--Yau fibrations over curves only with fixed volumes of general fibers and Iitaka volumes. As its application, we construct a separated coarse moduli space of K-stable Calabi-Yau fibrations over curves…

We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples,…

We show that a polarised manifold with a constant scalar curvature K\"ahler metric and discrete automorphisms is K-stable. This refines the K-semistability proved by S. K. Donaldson.

We develop a moduli theory of algebraic varieties and pairs of non-negative Kodaira dimension. We define stable minimal models and construct their projective coarse moduli spaces under certain natural conditions. This can be applied to a…

We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties.

In this thesis we study the relationship between the existence of canonical metrics on a complex manifold and stability in the sense of geometric invariant theory. We introduce a modification of K-stability of a polarised variety which we…

K-polystability of a polarised variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature K\"ahler metric. When a variety is K-unstable, it is expected to admit a "most destabilising"…

We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties which should be put on the boundary of its…

Let k be an algebraically closed field of characteristic zero. Let f:X-->S be a flat, projective morphism of k-schemes of finite type with integral geometric fibers. We prove existence of a projective relative moduli space for semistable…

We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we…

We construct a moduli space of polarised manifolds which admit a constant scalar curvature K\"ahler metric. We show that this space admits a natural K\"ahler metric.

We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact…