arXiv research

Reductive quotients preserve klt singularities in algebraic geometry.

Surveying stability of klt singularities with new solutions.

Study projective KLT varieties with projectively flat cotangent sheaves.

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

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

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

Uniformizes klt pairs using bounded symmetric domains.

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…

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

The paper classifies certain singular projective varieties with specific properties.

Proves finitely generated graded rings for klt singularities.

Criterion for projectivisation on klt spaces, characterizing quotients and stability.

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

KLT picker automates particle picking for cryo-EM, especially for low SNR images.

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

Paper extends Hodge correspondence to singular Kähler spaces.

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…

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

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…

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

The paper classifies minimal projective varieties satisfying a specific equality.

Characterizes stable sheaves for equality in orbifold BG inequality.

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

Study on positivity of CM line bundles on moduli space of klt good minimal models with κ=1.

Establishes Hermite-Einstein metrics on complex spaces with singularities.

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 paper proves ACC for local volumes under boundedness conditions.

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…

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

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…

Optimizes bounds for threefold singularity volumes.

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…

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…

The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.

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…

For any flat projective family $(\mX,\mL)\rightarrow C$ such that the generic fibre $\mX_η$ is a klt Q-Fano variety and $\mL|_{\mX_η}\sim_{Q}-K_{X_η}$, we use the techniques from the minimal model program (MMP) to modify the total family. The end product is a family such that every fiber is a klt Q-Fano variety. Moreov…

In this paper, we study a projective klt pair $(X, Δ)$ with the nef anti-log canonical divisor $-(K_X+Δ)$ and its maximally rationally connected fibration $ψ: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+Δ)$ on $X$ coincides with that of the anti-log canonical div…

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

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…

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

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

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

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

Proof of complex geometry theorem for specific singular spaces.

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

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…