63 citations · 275 across the 28 of their papers we have counts for
30 papers · 1 filter
Stable Degeneration, Non-degenerate Forms, and Kaledin's Conjecture
Chenyang Xu, Ziquan Zhuang, Henri Guenancia
We prove that stable degeneration, the canonical degeneration associated to the normalized volume minimizer of a Kawamata log terminal (klt) singularity, preserves non-degenerate r…
Open problems in K-stability of Fano varieties
Chenyang Xu, Ziquan Zhuang
In this note, we discuss a number of open problems in K-stability theory.
Relative stability theory and properness of K-moduli spaces
Harold Blum, Yuchen Liu, Chenyang Xu +1
We define the relative stability threshold of a family of Fano varieties over a DVR and show that it is computed by a divisorial valuation. In the case when the special fiber is K-…
Boundedness of log Fano cone singularities and discreteness of local volumes
Chenyang Xu, Ziquan Zhuang
We prove that in any fixed dimension, K-semistable log Fano cone singularities whose volumes are bounded from below by a fixed positive number form a bounded set. As a consequence,…
Stability of klt singularities
Ziquan Zhuang
We survey some recent development in the stability theory of klt singularities. The main focus is on the solution of the stable degeneration conjecture.
Birational rigidity and alpha invariants of Fano varieties
Ivan Cheltsov, Arman Sarikyan, Ziquan Zhuang
We prove that for every , there is a birationally super-rigid Fano variety such that . Also we show that for every ,…
On boundedness of singularities and minimal log discrepancies of Kollár components, II
Ziquan Zhuang
We show that a set of K-semistable log Fano cone singularities is bounded if and only if their local volumes are bounded away from zero, and their minimal log discrepancies of Koll…
Direct summands of klt singularities
Ziquan Zhuang
We show that direct summands (or more generally, pure images) of klt type singularities are of klt type. As a consequence, we give a different proof of a recent result of Braun, Gr…
Stable degenerations of singularities
Chenyang Xu, Ziquan Zhuang
For any Kawamata log terminal (klt) singularity and any minimizer of its normalized volume function, we prove that the associated graded ring is always finitely generated, as conje…
On boundedness of singularities and minimal log discrepancies of Kollár components
Ziquan Zhuang
Recent study in K-stability suggests that klt singularities whose local volumes are bounded away from zero should be bounded up to special degeneration. We show that this is true i…
The existence of the Kähler-Ricci soliton degeneration
Harold Blum, Yuchen Liu, Chenyang Xu +1
We prove an algebraic version of the Hamilton-Tian Conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a r…
Finite generation for valuations computing stability thresholds and applications to K-stability
Yuchen Liu, Chenyang Xu, Ziquan Zhuang
We prove that on any log Fano pair of dimension whose stability threshold is less than , any valuation computing the stability threshold has a finitely generated…
Seshadri constants and K-stability of Fano manifolds
Hamid Abban, Ziquan Zhuang
We give a lower bound of the -invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of F…
Uniqueness of the minimizer of the normalized volume function
Chenyang Xu, Ziquan Zhuang
We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. This is achieved by defining stab…
Optimal destabilizing centers and equivariant K-stability
Ziquan Zhuang
We give an algebraic proof of the equivalence of equivariant K-semistability (resp. equivariant K-polystability) with geometric K-semistability (resp. geometric K-polystability). A…
K-stability of Fano varieties via admissible flags
Hamid Abban, Ziquan Zhuang
We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces…
On positivity of the CM line bundle on K-moduli spaces
Chenyang Xu, Ziquan Zhuang
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 p…
Some criteria for uniform K-stability
Chuyu Zhou, Ziquan Zhuang
We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to -invariant having a positive lower bound. Then…
On the Cheltsov--Rubinstein conjecture
Kento Fujita, Yuchen Liu, Hendrik Süß +2
In this note we investigate the Cheltsov--Rubinstein conjecture. We show that this conjecture does not hold in general and some counterexamples will be presented.
Product theorem for K-stability
Ziquan Zhuang
We prove a product formula for -invariant and as an application, we show that product of K-(semi, poly)stable Fano varieties is also K-(semi, poly)stable.
On the sharpness of Tian's criterion for K-stability
Yuchen Liu, Ziquan Zhuang
Tian's criterion for K-stability states that a Fano variety of dimension whose alpha invariant is greater than is K-stable. We show that this criterion is sharp…
Birational superrigidity is not a locally closed property
Ziquan Zhuang
We prove an optimal result on the birational rigidity and K-stability of index hypersurfaces in with ordinary singularities when and also study the…
Birational superrigidity and K-stability of singular Fano complete intersections
Yuchen Liu, Ziquan Zhuang
We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information fr…
Birational superrigidity and K-stability of Fano complete intersections of index one (with an appendix written jointly with Charlie Stibitz)
Ziquan Zhuang
We prove that every smooth Fano complete intersection of index and codimension in is birationally superrigid and K-stable if . We also propose…