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Ziquan Zhuang

Publications and source records attributed to Ziquan Zhuang.

At least 19 recordsLinked to original sources
arXiv2026

Stable Degeneration, Non-degenerate Forms, and Kaledin's Conjecture

We prove that stable degeneration, the canonical degeneration associated to the normalized volume minimizer of a Kawamata log terminal (klt) singularity, preserves non-degenerate reflexive differential forms. In particular, the stable degeneration of a symplectic singularity is again symplectic. Combining this with a deformation-theoretic rigidity result for symplectic degenerations, we confirm Kaledin's conjecture that the formal completion of any symplectic singularity is conical. As applications, we show that the natural base of any normalized nilpotent orbit closure is a K-semistable Fano variety, and that the normalized volume minimizer of a hypertoric singularity is induced by the standard dilation.

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arXiv2025

Relative stability theory and properness of K-moduli spaces

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-unstable, but the generic fiber is K-semistable, we use the divisorial valuation computing the threshold to replace the special fiber by a new one with a strictly larger stability threshold. Iterating this process yields a new and more direct proof of the properness of the K-moduli space that uses only birational geometry arguments.

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arXiv2025

Boundedness in general type MMP

We show that in any sequence of a general type MMP, the minimal log discrepancy of singularities takes at most finitely many values, and the fibers of all the extremal contractions and flips belong to a bounded family. A key ingredient in the proof is an analysis of the behavior of local volumes in the MMP.

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arXiv2025

Essential dimension of isogenies

We give a lower bound for the essential dimension of isogenies of complex abelian varieties. The bound is sharp in many cases. In particular, the multiplication-by-$m$ map is incompressible for every $m\geq 2$, confirming a conjecture of Brosnan.

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arXiv2025

Non-existence of Ulrich modules over Cohen-Macaulay local rings

Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal.

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arXiv2025

Stable degenerations of singularities

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 conjectured by Chi Li. As a consequence, we complete the last step of establishing the Stable Degeneration Conjecture proposed by Chi Li and the first named author for an arbitrary klt singularity.

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arXiv2024

Direct summands of klt singularities

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, Greb, Langlois and Moraga that reductive quotients of klt type singularities are of klt type.

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arXiv2024

Boundedness of log Fano cone singularities and discreteness of local volumes

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, we show that the set of local volumes of klt singularities of a fixed dimension has zero as the only accumulation point.

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arXiv2023

On boundedness of singularities and minimal log discrepancies of Kollár components, II

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 Kollár components are bounded from above. As corollaries, we confirm the boundedness conjecture for K-semistable log Fano cone singularities in dimension three, and show that local volumes of 3-dimensional klt singularities only accumulate at zero.

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arXiv2023

MMP for locally stable families and wall crossing for moduli of stable pairs

We construct reduction and wall-crossing morphisms between the moduli spaces of stable pairs as the coefficients vary, generalizing the earlier work of Ascher, Bejleri, Inchiostro and Patakfalvi which deals with the klt case. Along the proof, we show that one can run the MMP with scaling on normal locally stable families over a normal base, and that the existence of good minimal models is preserved when reducing coefficients away from zero.

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arXiv2023

Stability of klt singularities

We survey some recent development in the stability theory of klt singularities. The main focus is on the solution of the stable degeneration conjecture.

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arXiv2023

The existence of the Kähler-Ricci soliton degeneration

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 reduced uniformly Ding stable triple, which admits a Kähler-Ricci soliton when the ground field $\mathbb{k}=\mathbb{C}$.

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arXiv2023

On boundedness of singularities and minimal log discrepancies of Kollár components

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 in dimension three, or when the minimal log discrepancies of Kollár components are bounded from above. We conjecture that the minimal log discrepancies of Kollár components are always bounded from above, and verify it in dimension three when the local volumes are bounded away from zero. We also answer a question of Han, Liu and Qi on the relation between log canonical thresholds and local volumes.

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arXiv2023

Birational rigidity and alpha invariants of Fano varieties

We prove that for every $ε>0$, there is a birationally super-rigid Fano variety $X$ such that $\frac{1}{2}\leqslantα(X)\leqslant \frac{1}{2}+ε$. Also we show that for every $ε>0$, there is a Fano variety $X$ and a finite subgroup $G\subset\mathrm{Aut}(X)$ such that $X$ is $G$-birationally super-rigid, and $α_G(X)<ε$.

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arXiv2022

K-stability of Fano varieties via admissible flags

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 of Fano index two, (b) to compute the stability thresholds for hypersurfaces at generalized Eckardt points and for cubic surfaces at all points, and (c) to provide a new algebraic proof of Tian's criterion for K-stability, amongst other applications.

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arXiv2022

Seshadri constants and K-stability of Fano manifolds

We give a lower bound of the $δ$-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.

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