# Nonexistence of asymptotic GIT compactification

```@article{Wang2014NonexistenceOA,
title={Nonexistence of asymptotic GIT compactification},
author={Xiaowei Wang and Xu Chen},
journal={Duke Mathematical Journal},
year={2014},
volume={163},
pages={2217-2241}
}```
• Published 2014
• Mathematics
• Duke Mathematical Journal
We provide examples of families of (log) smooth canonically polarized varieties, including smooth weighted pointed curves and smooth hypersurfaces in \$P^3\$ with large degree such that the Chow semistable limits under distinct pluricanonical embeddings do not stabilize.
24 Citations
Boundedness of moduli of varieties of general type
• Mathematics
• 2014
We show that the family of semi log canonical pairs with ample log canonical class and with fixed volume is bounded.
Separatedness of moduli of K-stable varieties
• Mathematics
• 2013
Given a one parameter flat family of polarized algebraic varieties, we show that any K-stable limit is unique. In particular, moduli spaces of K-stable polarized varieties are automatically HausdorffExpand
Logarithmic Chow semistability of polarized toric manifolds
The logarithmic Chow semistability is a notion of Geometric Invariant Theory for the pair consists of varieties and its divisors. In this paper we introduce a obstruction of semistability forExpand
On the projectivity of the moduli space of stable surfaces in characteristic p>5
We prove that every proper subspace of the moduli space of stable surfaces with fixed volume over an algebraically closed field of characteristic p>5 is projective. As a consequence we also deduceExpand
Minimising CM degree and slope stability of projective varieties
We discuss a minimization problem of the degree of the CM line bundle among all possible fillings of a polarized family with fixed general fibers. We show that such minimization implies the slopeExpand
Ampleness of the CM line bundle on the moduli space of canonically polarized varieties
• Mathematics, Physics
• 2015
We prove that the CM line bundle is ample on the proper moduli space which parametrizes KSBA stable varieties.
GIT Stability, K-Stability and the Moduli Space of Fano Varieties
This is a slightly extended version of the lecture notes of a mini-course in the workshop of Moduli of K-stable Varieties given by the author, in which the main construction of the proper moduliExpand
On the moduli of Kähler-Einstein Fano manifolds
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 varietiesExpand
Uniqueness of K-polystable degenerations of Fano varieties.
• Mathematics
• 2018
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 latterExpand
On the proper moduli spaces of smoothable Kähler–Einstein Fano varieties
• Mathematics
• Duke Mathematical Journal
• 2019
In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano K\"ahler-Einstein manifolds inside an appropriate Hilbert scheme. InExpand

#### References

SHOWING 1-10 OF 49 REFERENCES
Birational Geometry of Algebraic Varieties
• Mathematics
• 1998
1. Rational curves and the canonical class 2. Introduction to minimal model program 3. Cone theorems 4. Surface singularities 5. Singularities of the minimal model program 6. Three dimensional flopsExpand
Existence of minimal models for varieties of log general type
• Mathematics
• 2006
Assuming finite generation in dimension n − 1, we prove that pl-flips exist in dimension n.
A simply connected surface of general type with pg = 0 and K2 = 4
• Mathematics
• 2009
As the sequel to [5, 7], we construct a simply connected minimal complex surface of general type with p_g = 0 and K^2 = 4 by using a rational blow-down surgery and Q-Gorenstein smoothing theory.
Simply connected surfaces of general type in positive characteristic via deformation theory
• Mathematics
• 2013
Algebraically simply connected surfaces of general type with p_g=q=0 and 1\le K^2\le 4 in positive characteristic (with one exception in K^2=4) are presented by using a Q-Gorenstein smoothing ofExpand
Existence of log canonical closures
• Mathematics
• 2011
Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U0⊂U, such that (X,Δ)×UU0 has a good minimal model over U0 and the images of all theExpand
Moduli spaces of weighted pointed stable curves
Abstract A weighted pointed curve consists of a nodal curve and a sequence of marked smooth points, each assigned a number between zero and one. A subset of the marked points may coincide if the sumExpand
Scalar Curvature and Projective Embeddings, I
We prove that a metric of constant scalar curvature on a polarised Kahler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introducedExpand
Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fibre Spaces
Let V and W be non-singular projective varieties over the field of complex numbers C, n= dim (V) and m=dim (W). Let/: V---+W be a fibre space (this simply means that I is surjective with connectedExpand
A study of the Hilbert-Mumford criterion for the stability of projective varieties
• Mathematics
• 2004
We make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties \$(X,L)\$; in particular for K- and Chow stability. For each type ofExpand