arXiv research

Surveying stability of klt singularities with new solutions.

Reductive quotients preserve klt singularities in algebraic geometry.

The minimizer of a volume function is unique for klt singularities.

Study projective KLT varieties with projectively flat cotangent sheaves.

Proves finitely generated graded rings for klt singularities.

Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.

The paper proves the existence of singular cscK metrics on smoothable varieties.

Paper extends Hodge correspondence to singular Kähler spaces.

Establishes Hermite-Einstein metrics on complex spaces with singularities.

Optimizes bounds for threefold singularity volumes.

Proves boundedness of log Fano cone singularities with bounded local volumes.

We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techni…

The paper classifies certain singular projective varieties with specific properties.

The paper proves conditions for the Abundance conjecture in minimal projective klt pairs.

Study shows boundedness of klt singularities in 3D or with bounded Kollár components.

The paper proves ACC for local volumes under boundedness conditions.

Complex projective varieties are quotients of polydiscs under specific group actions.

We generalise Simpson's nonabelian Hodge correspondence to the context of projective varieties with klt singularities. The proof relies on a descent theorem for numerically flat vector bundles along birational morphisms. In its simplest form, this theorem asserts that given any klt variety X and any resolution of singu…

Characterizes stable sheaves for equality in orbifold BG inequality.

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistabili…

We show that in any $\mathbb{Q}$-Gorenstein flat family of klt singularities, normalized volumes can only jump down at countably many subvarieties. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed…

Let (X,D) be a klt pair. Assuming either K_X+D big or -(K_X+D) ample, and that the coefficients of D are greater than 1/2, we show that the Kähler-Einstein metric attached to (X,D) -whenever it exists- has cone singularities along D on the log-smooth locus of the pair intersected with the ample locus of K_X+D (in the n…

The Chow-Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements abou…

The paper proves a key inequality for a specific type of complex spaces.

This is a survey on the recent theory on minimizing the normalized volume function attached to any klt singularities.

We prove that among all Kollár components obtained by plt blow ups of a klt singularity $o \in (X, D)$, there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Kollár component exists, it uniquely minimizes the normalized volume function introduced in [Li15a] among all divisorial valu…

Given a klt singularity $x\in (X, D)$, we show that a quasi-monomial valuation $v$ with a finitely generated associated graded ring is the minimizer of the normalized volume function $\widehat{\rm vol}_{(X,D),x}$, if and only if $v$ induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a mi…

Proof of complex geometry theorem for specific singular spaces.

The paper generalizes K-stability results to singular and weighted settings.

We investigate the holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor. Finiteness of the number of connected components, a Bochner principle for holomorphic tensors, and a connection between irreducibility of holonomy representations and stability of the tange…

We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball…

The Miyaoka-Yau inequality is proven for certain singular varieties with big canonical or anticanonical divisors.

Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.

We study several questions involving relative Ricci-flat Kähler metrics for families of log Calabi-Yau manifolds. Our main result states that if $p:(X,B)\to Y$ is a Kähler fiber space such that $\displaystyle (X_y, B|_{X_y})$ is generically klt, $K_{X/Y}+B$ is relatively trivial and $p_*(m(K_{X/Y}+B))$ is Hermitian fla…

Let $X$ be a normal compact Kähler space with klt singularities and torsion canonical bundle. We show that $X$ admits arbitrarily small deformations that are projective varieties if its locally trivial deformation space is smooth. We then prove that this unobstructedness assumption holds in at least three cases: if $X$…

Study confirms boundedness of certain singularities in log Fano geometry.

We establish the existence of the K"ahler-Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song-Tian \cite{ST3} in case of projective varieties with klt singularities. We also prove that the normalized K"ahler-Ricci flow will converge to the \ka-Eins…

New invariants prove existence of Kahler-Einstein metrics on big classes.

Extends K-stability theory to projective klt pairs with a big anticanonical class.

The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.

The study shows how certain singular Kähler metrics can define Kähler currents and RCD spaces.

For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, we introduce a normalized volume function $\widehat{\rm vol}$ that is defined on the space of real valuations centered at $o$ and consider the problem of minimizing $\widehat{\rm vol}$. We prove that the normalized volume has a uniform positive lower bound by pro…

The study of projective varieties with nef anticanonical divisors and log terminal singularities.

Decomposes singular Kähler spaces with trivial first Chern class into simpler components.

Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.

Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.

Uniformizes klt pairs using bounded symmetric domains.

The paper studies algebraic fibre spaces with specific properties and proves key results about their structure.