Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
Reductive quotients preserve klt singularities in algebraic geometry.
problem Preserving klt singularities in quotients of klt singularities.
method Proving that the quotient of a klt type singularity by a reductive group is of klt type.
result The quotient of a klt variety by a reductive group results in a klt variety with a suitable boundary.
The paper classifies certain singular projective varieties with specific properties.
problem Classifying projective klt pairs with nef anti-log canonical divisors.
method Establishes a structure theorem using locally trivial rationally connected fibrations.
result Projective klt pairs can be decomposed into rationally connected and Calabi-Yau varieties.
Generalizes Toledo invariant to singular klt varieties, proving Milnor-Wood inequality.
problem Proving Milnor-Wood inequality for singular klt varieties.
method Generalized Toledo invariant to singular klt varieties and proved Milnor-Wood inequality.
result Milnor-Wood inequality for singular klt varieties.
Proves CM line bundle positivity for K-stable klt Fano varieties.
problem Proving ampleness of the CM line bundle for K-stable klt Fano varieties.
method Algebraic approach, including probability theory for limit computations.
result Proves semi-positivity and positivity statements for K-semi-stable and uniform K-stable cases.
Generalizes nonabelian Hodge theory to klt singularities.
problem Applying nonabelian Hodge theory to spaces with klt singularities.
method Uses descent theorems for numerically flat vector bundles and a new restriction theorem for semistable Higgs sheaves.
result Establishes a new restriction theorem for semistable Higgs sheaves.
We prove a conjecture about log canonical thresholds and volumes of klt singularities.
problem Understanding the log canonical thresholds and volumes of klt singularities.
method We use quasi-monomial valuations and techniques from klt singularities to prove the conjectures.
result We confirm Chi Li's conjecture and show that the volume of klt singularities is a constructible function.
Uniformizes varieties of general type with harmonic metrics.
problem Uniformizing varieties of general type with harmonic metrics.
method Criterion for harmonic metrics on Higgs bundles on klt varieties.
result Complete numerical characterisation of singular quotients of the unit ball.
Study on Klt varieties with trivial canonical class, focusing on holonomy and stability.
problem Holonomy group of singular Kähler-Einstein metrics on klt varieties with numerically trivial canonical divisor.
method Investigation of holonomy group properties, finiteness of connected components, Bochner principle for holomorphic tensors, and connections between irreducibility of holonomy representations and stability of the tangent sheaf.
result Refinement of known decompositions for tangent sheaves of varieties with trivial canonical divisor, showing that up to finite quasi-étale covers, varieties with strongly stable tangent sheaves are either Calabi-Yau or irreducible holomorphic symplectic.
The paper proves the existence of singular cscK metrics on smoothable varieties.
problem Existence of singular cscK metrics on smoothable varieties.
method Developing a strong topology of pluripotential theory in families and uniform estimates for cscK metrics.
result Existence of singular cscK metrics on Q-Gorenstein smoothable klt varieties when the Mabuchi functional is coercive. The paper classifies minimal projective varieties satisfying a specific equality.
problem Classifying minimal projective varieties with a specific equality.
method Established a structure theorem for minimal projective klt varieties satisfying Miyaoka's equality.
result Minimal projective klt varieties with Miyaoka's equality have semi-ample canonical divisors and specific Kodaira dimensions.
Criterion for projectivisation on klt spaces, characterizing quotients and stability.
problem Characterizing finite quotients of projective spaces and Abelian varieties.
method Criterion based on reflexive sheaves and stability conditions.
result Characterization of finite quotients using Q-Chern class inequalities and stability condition. Extends K-stability theory to projective klt pairs with a big anticanonical class.
problem Behavioral pathologies in K-stability for projective klt pairs with a big anticanonical class.
method Extends K-stability theory to projective klt pairs with a big anticanonical class, observing that K-semistability forces a klt anticanonical model with the same stability property.
result K-semistability forces projective klt pairs with a big anticanonical class to have a klt anticanonical model with the same stability property.
Complex projective varieties are quotients of polydiscs under specific group actions.
problem Characterizing complex projective varieties as quotients of polydiscs.
method Proving varieties are quotients by groups acting properly discontinuously and freely in codimension one.
result Complex projective varieties with klt singularities and ample canonical divisors are quotients of the polydisc.
Characterizes stable sheaves for equality in orbifold BG inequality.
problem Stability of sheaves on compact Kähler varieties with klt singularities.
method Characterization of stable reflexive sheaves for BG equality.
result Characterizes stable reflexive sheaves for equality in BG inequality.
The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.
problem Understanding the structure of projective klt varieties with nef tangent sheaves.
method Developing theory of positivity of coherent sheaves and proving structure theorem.
result Projective klt varieties with specific tangent sheaf properties admit rationally connected fibrations onto abelian varieties.
Paper extends Hodge correspondence to singular Kähler spaces.
problem Establishing Hodge correspondence over Kähler spaces with singularities.
method Using equivalence of polystable Higgs bundles and semi-simple flat bundles over regular loci, and descent theorem for semistable Higgs bundles.
result Non-abelian Hodge correspondence established over compact Kähler klt spaces and their regular loci.
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…
Establishes Hermite-Einstein metrics on complex spaces with singularities.
problem Existence of Hermite-Einstein metrics on complex spaces with singularities.
method Established existence of estimable Hermite-Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces with klt singularities.
result Obtained precise results for varieties with klt singularities.
The paper studies fundamental groups of compact Kahler varieties with nef anti-canonical bundles.
problem Understanding the fundamental groups of compact Kahler varieties with specific properties.
method Application of geometric analytic theory of Kahler spaces and study of Albanese maps.
result The fundamental group of compact Kahler varieties with nef anti-canonical bundle is almost Abelian.
The Miyaoka-Yau inequality is proven for certain singular varieties with big canonical or anticanonical divisors.
problem Establishing the Miyaoka-Yau inequality for singular varieties with specific divisors.
method Defining the non-pluripolar product and establishing the Bogomolov-Gieseker type inequality for Higgs sheaves; investigating second Chern class inequalities.
result Proven the Miyaoka-Yau inequality for projective klt varieties with big canonical or anticanonical divisors.
Proof of complex geometry theorem for specific singular spaces.
problem Proving a complex geometry theorem for a specific type of singular spaces.
method Self-contained proof of singular Beauville-Bogomolov decomposition theorem.
result Proof of singular Beauville-Bogomolov decomposition theorem for compact Kähler varieties with log terminal singularities and zero first Chern class.
The paper generalizes K-stability results to singular and weighted settings.
problem Generalizing K-stability to singular and weighted settings.
method Generalization of results in \cite{Li22a} to singular and weighted settings.
result The \(\mathbb{G}\)-uniform weighted K-stability for models implies \(\mathbb{G}\)-coercivity of the weighted Mabuchi functional.
The paper shows how certain complex projective varieties can be broken down into simpler types.
problem Understanding the structure of complex projective varieties with pseudo-effective tangent sheaves.
method Developed a theory of pseudo-effective sheaves and applied the minimal model program.
result Projective klt varieties with pseudo-effective tangent sheaves can be decomposed into Fano varieties and Q-abelian varieties.
The study of projective varieties with nef anticanonical divisors and log terminal singularities.
problem Understanding the structure and properties of projective varieties with specific divisor conditions.
method Analyzing the Albanese map and MRC fibration for klt projective varieties, showing locally constant fibrations and product decompositions.
result Generalization of results for smooth projective varieties to the klt case, including decomposition into rationally connected and projective varieties with trivial canonical divisor.
Equivalence proven between uniformizing varieties and tensors, generalizing uniformization results.
problem Characterizing complex-projective varieties with klt singularities and ample canonical divisors.
method Constructing a uniformizing variation of Hodge structure from slope zero tensors and vice versa.
result Generalization of uniformization results to singular settings, including quotients of tube domains.
Shows CM line bundles are ample on K-stable varieties.
problem Ensuring CM line bundles are ample on K-stable varieties.
method Analyzes CM line bundles on K-stable varieties and their families.
result CM line bundles are ample on K-stable varieties with maximal variation.
The paper studies algebraic fibre spaces with specific properties and proves key results about their structure.
problem Analyzing properties of algebraic fibre spaces with strictly nef relative anti-log canonical divisors.
method Using projective klt pairs and fibration techniques, the paper proves locally constant fibration properties and rational connectedness.
result The fibration is locally constant with rationally connected fibers, and the base is a canonically polarized hyperbolic projective manifold.
Study projective klt pairs with nef anti-canonical divisor and their properties.
problem Characterize properties of projective klt pairs with nef anti-log canonical divisors.
method Analyze maximally rationally connected fibration and use numerical dimension analogy.
result Numerical dimension of anti-log canonical divisor on X matches that on a general fiber. New invariants prove existence of Kahler-Einstein metrics on big classes.
problem Existence of Kahler-Einstein metrics on varieties with klt singularities.
method Introducing new invariants and proving a generalization of the Tian-Odaka-Sano Theorem.
result Proves existence of twisted Kahler-Einstein metrics on big classes.
Surveying stability of klt singularities with new solutions.
problem Stability of klt singularities.
method Survey and solution of the stable degeneration conjecture.
result Solution to the stable degeneration conjecture.
Study Kähler-Einstein potentials on stable varieties near singularities
problem Asymptotic behavior of Kähler-Einstein potentials on stable varieties near singularities
method Using iterated logarithmic functions and refined lower bounds
result Improved estimates for Kähler-Einstein potentials
Establishes K"ahler-Ricci flow on log canonical varieties.
problem Existence and convergence of K"ahler-Ricci flow on varieties with log canonical singularities.
method Generalizes previous results for klt singularities, proves convergence, and constructs solutions with flips.
result Existence and convergence of K"ahler-Ricci flow on semi-log canonical models.
The minimizer of a volume function is unique for klt singularities.
problem Uniqueness of the minimizer of the normalized volume function for klt singularities.
method Defining stability thresholds for valuations and showing K-semistability.
result The minimizer of the normalized volume function for a klt singularity is unique up to rescaling.
Equality in Miyaoka-Yau inequality implies uniformization of Klt pairs.
problem Understanding uniformization of Klt pairs under equality in Miyaoka-Yau inequality.
method Analyzing Kähler klt pairs with specific conditions and using orbifold Miyaoka-Yau inequality.
result Orbifold universal cover is either the unit ball or affine space.
The paper addresses deformations of Kähler spaces with vanishing first Chern class.
problem Deformations of Kähler spaces with specific properties.
method Analyzes locally trivial deformation spaces and uses cohomological vanishing conditions.
result Shows that under certain conditions, deformations of Kähler spaces are projective varieties.
Uniformizes klt pairs using bounded symmetric domains.
problem Characterizing klt pairs uniformizable by bounded symmetric domains.
method Determines conditions for uniformization using Miyaoka-Yau-type inequalities.
result Characterizations of orbifold quotients of polydisc and classical bounded symmetric domains.
Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.
problem Understanding geometric structures of singular Kahler spaces.
method Proving RCD spaces homeomorphic to 3D projective varieties with bounded Nash entropy and Ricci curvature.
result Compact RCD spaces are equivalent to underlying projective varieties.
The normalized volume is lower semicontinuous in klt singularities.
problem Lower semicontinuity of normalized volumes in klt singularities.
method Flat family of klt singularities, alternative characterization of K-semistability.
result K-semistability is very generic or empty in log Fano pairs.
Two types of nonvanishing results are presented for compact Kähler varieties.
problem Deriving geometric consequences from numerical information in Kähler geometry.
method Analyzing non-uniruled varieties and hyperkähler manifolds to establish nonvanishing results for adjoint and nef bundles.
result Strong abundance-type results are obtained in dimension 4.
Proves finitely generated graded rings for klt singularities.
problem Understanding the structure of klt singularities.
method Analyzes graded rings associated with minimizers of normalized volume functions.
result Graded rings are finitely generated for klt singularities.
The paper proves a unique Kollár component is K-semistable among plt blow ups of a klt singularity.
problem Stability of valuations and Kollár components in klt singularities.
method Proving minimization of the normalized volume function yields K-semistable Kollár components.
result There is at most one Kollár component that is (log-)K-semistable among plt blow ups of a klt singularity.
Survey on minimizing a volume function for singularities.
problem Minimizing the normalized volume function for klt singularities.
method Theoretical survey and analysis of recent research.
result Survey of recent findings on minimizing the normalized volume function.
The paper proves a key inequality for a specific type of complex spaces.
problem Establishing a mathematical inequality for a class of complex spaces.
method Analytical approach involving Higgs sheaves and orbifolds.
result Proves the Miyaoka-Yau inequality for minimal Kähler klt spaces.
Study thresholds and K-stability using valuations in complex projective varieties.
problem Understanding thresholds and K-stability in complex projective varieties.
method Using valuations to study log canonical and stability thresholds, proving infima are attained for ample L, and obtaining simple expressions in toric cases.
result The thresholds can be written as infima of functionals on the space of valuations, and are attained for ample L.
The paper establishes a Miyaoka-Yau inequality for Fano varieties and their equivariant versions.
problem Investigating the Miyaoka-Yau inequality for Fano varieties and their equivariant versions.
method Using delta invariant and equivariant Chern classes, the paper formulates and proves an equivariant Miyaoka-Yau inequality for smooth Fano manifolds with Kähler-Ricci solitons.
result The paper proves the Miyaoka-Yau inequality for Fano varieties and their equivariant versions.
Finite degree formula proven for normalized volumes of klt singularities.
problem Proving a finite degree formula for normalized volumes of klt singularities.
method Using finite surjective morphisms and properties of klt singularities.
result Normalized volumes satisfy a finite degree formula.
KLT picker automates particle picking for cryo-EM, especially for low SNR images.
problem Challenges in particle picking, especially for low SNR micrographs.
method Data-driven optimal templates learned via Karhunen Loeve Transform.
result High-quality results with minimal manual effort on publicly available data sets.