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.
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.
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.
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.
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 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.
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.
Study shows convergence of anticanonically balanced metrics to Kähler-Einstein metrics on Fano manifolds.
problem Finding anticanonically balanced metrics on Fano manifolds.
method Simplification of Donaldson's proof using Berezin-Toeplitz quantization.
result Sequence of anticanonically balanced metrics converges to Kähler-Einstein metric.
In this paper we provide a classification of all Moishezon twistor spaces on the connected sum of four complex projective planes. This is given by means of the anticanonical system of the twistor spaces. In particular, we show that the anticanonical map is birational, two to one over the image, or otherwise the image o…
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.
Fundamental groups of certain Kähler orbifolds have polynomial growth.
problem Understanding the fundamental groups of specific types of orbifolds.
method Analyzing the orbifold fundamental group with respect to the nef anticanonical bundle.
result The orbifold fundamental group has polynomial growth.
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.
Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.
problem Existence of Kähler-Einstein metrics on manifolds with big anticanonical class.
method Developed a theory of Deligne functionals and slope formulas for singular metrics, proving a slope formula for the Ding functional in the big setting.
result Existence of a unique Kähler-Einstein metric implies uniform Ding stability.
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.
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.
New Calabi-Yau metrics converge polynomially to Calabi model space.
problem Finding complete Calabi-Yau metrics with polynomial convergence rate.
method Defined new metrics on Calabi-Yau complements with ample normal bundles.
result Uniqueness of these metrics within a cohomology class.
Study the Albanese map for Kähler manifolds with nef anticanonical bundle.
problem Characterize the structure of the Albanese map for Kähler manifolds with nef anticanonical bundle.
method Analyze two cases: general fiber is Calabi-Yau or projective space. Provide proofs for both cases.
result For the case where the general fiber is Calabi-Yau, the manifold itself must be Calabi-Yau.
Study on positivity of CM line bundles on moduli space of klt good minimal models with κ=1.
problem Positivity of CM line bundles on moduli space of klt good minimal models with κ=1.
method Construction of a moduli space of numerical equivalence classes, proving projectivity of moduli space of ε-stable quotients, and using K-moduli of quasimaps.
result CM line bundle becomes ample after normalization and moduli space is quasi-projective.
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.
We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical Q-divisors. First, we propose a condition in terms of certain anticanonical Q-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that …
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. 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.
In this paper we classify all Moishezon twistor spaces on 4CP^2. The classification is given in terms of the structure of the anticanonical system of the twistor spaces. We show that the anticanonical map satisfies one of the following three properties: (a) birational over the image, (b) two to one over the image, or (…
New complete Calabi-Yau metrics found in complex space.
problem Finding metrics on complex spaces with specific conditions.
method Generalized Calabi ansatz, non-archimedean Monge-Ampère equation.
result Complete Calabi-Yau metrics constructed in Fano manifolds.
New stability criterion for Fano manifolds using anticanonically balanced metrics.
problem Stability conditions for Fano manifolds and their invariant δm. method Proof of equivalence between stability condition and anticanonically balanced metrics.
result Established a Hilbert-Mumford type criterion for δm>1. The paper proves conditions for the Abundance conjecture in minimal projective klt pairs.
problem Proving the Abundance conjecture for minimal klt pairs with non-zero canonical bundle.
method Analyzing asymptotic behavior of multiplier ideals and properties of supercanonical currents.
result Supercanonical currents are central to proving the Abundance conjecture.
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.
We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.
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.
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 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.
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.
Particle picking is currently a critical step in the cryo-EM single particle reconstruction pipeline. Despite extensive work on this problem, for many data sets it is still challenging, especially for low SNR micrographs. We present the KLT (Karhunen Loeve Transform) picker, which is fully automatic and requires as an …
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.
We continue to study twistor spaces on the connected sum of four complex projective planes, whose anticanonical map is of degree two over the image. In particular, we determine the defining equation of the branch divisor of the anticanonical map in an explicit form. Together with previous two articles (arXiv:1009.3153 …
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.
Study shows boundedness of klt singularities in 3D or with bounded Kollár components.
problem Boundedness of klt singularities in algebraic geometry.
method Analysis of Kollár components and local volumes.
result Minimal log discrepancies of Kollár components are bounded in dimension 3.
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.
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.
problem Proving the ACC conjecture for local volumes of klt singularities.
method Analyzing klt singularities with bounded ambient germs.
result ACC conjecture for local volumes holds under bounded conditions.
We show that in any 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 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.
problem Understanding the boundedness of log Fano cone singularities.
method Analyzes K-semistable log Fano cone singularities with bounded volumes.
result The set of local volumes of klt singularities has zero as the only accumulation point.
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…
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…
We study the evolution of anticanonical line bundles along the Kähler Ricci flow. We show that under some conditions, the convergence of Kähler Ricci flow is determined by the properties of the anticanonical divisors of M. As examples, the Kähler Ricci flow on M converges when M is a Fano surface and c12(M)=1…
Optimizes bounds for threefold singularity volumes.
problem Bounding local volumes of threefold singularities.
method Analyzes Gorenstein canonical non-hypersurface threefold singularities.
result Establishes optimal upper bound for local volumes.