Reductive automorphism groups for K-polystable Fano pairs proved.
problem Proving reductivity of automorphism groups for K-polystable Fano varieties.
method Establishing S-completeness and Θ-reductivity for moduli of K-semistable log Fano pairs, assuming K-semistability is open.
result K-polystable log Fano pairs have reductive automorphism groups.
The moduli continuity method is applied to describe K-polystable log Fano pairs.
problem Describing compact moduli spaces of K-polystable log Fano pairs.
method Applying the moduli continuity method to the log setting.
result Explicit construction of moduli spaces and ampleness of CM line bundle.
Finite group action on K-stability results in standard stability.
problem Understanding K-stability under finite group action.
method Analyzing G-equivariant K-semistability and K-polystability for log Fano pairs.
result G-equivariant K-semistability implies K-semistability for log Fano pairs.
Equivalence proven between algebraic stability and geometric stability.
problem Equivalence of algebraic and geometric stability criteria.
method Algebraic proof of equivalence, existence and uniqueness of minimal centers.
result Existence and uniqueness of minimal optimal destabilizing centers.
Proves new results on K-polystability of Q-Fano varieties.
problem K-polystability of Q-Fano varieties.
method Algebraic and differential-geometric arguments.
result Metric tangent cones of K-semistable log Fano cones are K-polystable.
Proves finitely generated associated graded rings for valuations on log Fano pairs.
problem Stability thresholds of log Fano pairs.
method Proves finite generation of associated graded rings for valuations.
result Log Fano pairs are uniformly K-stable if their stability threshold is less than a certain value.
Proves unique degeneration of log Fano fibration germs.
problem Stable degeneration of log Fano fibration germs.
method Introduced the H-invariant for filtrations over log Fano fibration germs and used a unique quasi-monomial valuation to achieve the degeneration.
result Unique K-polystable special degeneration of log Fano fibration germs.
Proves Yau-Tian-Donaldson conjecture for certain singular Fano varieties.
problem Proving the conjecture for a specific class of singular Fano varieties.
method Using log smooth resolutions and K-polystability criteria.
result Kähler-Einstein metrics exist for K-polystable singular Fano varieties.
Researchers classify K-stability of log Fano hyperplane arrangements.
problem Determining K-stability of log Fano hyperplane arrangements.
method Comprehensive analysis of K-stability conditions.
result Classification of K-stability for log Fano hyperplane arrangements.
New examples of Kähler-Ricci solitons on Fano threefolds with non-trivial moduli found.
problem Finding Fano threefolds with Kähler-Ricci solitons and non-trivial moduli.
method Established weighted K-stability and GIT-stability criteria, generalized Koiso's theorem, and developed the weighted Abban-Zhuang estimate.
result First examples of strictly weighted K-semistable Fano varieties and new examples of KRS Fano varieties with non-trivial moduli and small automorphism groups.
The paper constructs K-moduli spaces for plane curves and describes wall crossings.
problem Constructing and understanding K-moduli spaces for plane curves.
method Constructing proper good moduli spaces and establishing wall-crossing framework.
result The first wall crossing of K-moduli spaces for plane curves of degree 4 is a weighted blow-up of Kirwan type.
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.
The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.
problem Investigating K-polystability on Fano 4-folds with Lefschetz defect at least 2.
method Examining 19 families of Fano 4-folds with Lefschetz defect 3 and 175 families with Lefschetz defect 2, proving K-polystability and instability.
result Exactly 5 out of 19 families of Fano 4-folds with Lefschetz defect 3 are K-polystable, and 5 out of 175 Casagrande-Druel Fano 4-folds with Lefschetz defect 2 are K-polystable.
Unique K-polystable degenerations for Q-Fano varieties proven.
problem Proving uniqueness of K-polystable degenerations of Q-Fano varieties.
method Analyzing K-polystability and using moduli stack properties.
result K-polystable degenerations of Q-Fano varieties are unique.
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.
Study uniform K-stability and its connection to log Fano pairs' plt blowups.
problem Testing uniform K-stability of log Fano pairs.
method Evaluate volume function invariants for all plt blowups.
result Establishes uniform K-stability criteria for log Fano pairs.
Proves results on K-stability using arcs and Mabuchi functional.
problem K-stability in Fano manifolds and uniform K-polystability.
method Arcs and numerical criterion for stability of pairs.
result Characterizes coercivity of Mabuchi functional in terms of K-polystability.
Alternative proof of semipositivity and nefness for K-semistable log-Fano pairs.
problem Semipositivity and nefness of Chow-Mumford line bundle for K-semistable log-Fano pairs.
method Alternative proof using families of K-semistable log-Fano pairs.
result Proof of semipositivity and nefness for K-semistable log-Fano pairs.
Survey on Kähler-Einstein metrics on smoothable Fano varieties.
problem Existence of Kähler-Einstein metrics on Fano varieties.
method Analyzing properties of Fano varieties and their moduli spaces.
result Construction of compact algebraic moduli spaces of K-polystable Fano varieties.
It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functiona…
Unique K-polystable degenerations for Fano varieties confirmed.
problem Algebraic uniqueness of Kähler-Ricci flow limits on Fano manifolds.
method Study of optimal degeneration problems via new functionals of real valuations.
result Confirm algebraic uniqueness of Kähler-Ricci flow limits on Fano manifolds.
Survey on Kähler-Einstein and weighted solitons on Fano manifolds.
problem Existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds.
method Generalization of algebraic conditions for K-polystability.
result Existence of coupled Kähler-Einstein metrics and weighted solitons is equivalent to algebraic conditions.
Proves algebraic version of Hamilton-Tian conjecture for log Fano pairs.
problem Existence of Kähler-Ricci soliton degeneration for log Fano pairs.
method Algebraic proof of two-step degeneration to uniformly Ding stable triple.
result Log Fano pairs admit Kähler-Ricci soliton when ground field is complex.
Proves properness of K-moduli spaces for Fano varieties.
problem Proving properness of moduli spaces of K-polystable Fano varieties.
method Algebraic approach, studying test configurations, constructing stratification.
result Proves properness under specific divisorial valuation condition.
Study shows polystability of tangent and canonical sheaves on Kähler-Einstein log Fano pairs.
problem Stability of tangent and canonical sheaves on Kähler-Einstein log Fano pairs.
method Analysis of adapted tangent and canonical sheaves under singular Kähler-Einstein metrics.
result Adapted tangent and canonical sheaves are polystable.
Proves K-polystability for Kähler-Ricci shrinkers with decaying curvature.
problem K-stability of Kähler-Ricci shrinkers with decaying curvature.
method Developed algebraic theory for Kähler-Ricci shrinkers and proved K-polystability.
result Existence of Kähler-Ricci shrinker metric implies K-polystability in decaying curvature case.
Sharpness of Tian's K-stability criterion demonstrated through specific examples.
problem Determining the exact conditions for K-stability in Fano varieties.
method Constructing specific Fano varieties to test Tian's criterion's limits.
result Tian's criterion is sharp; there are Fano varieties meeting the criterion that are not K-polystable.
The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.
problem Finding Kähler-Einstein metrics on specific Fano varieties.
method Using combinatorial criteria for K-polystability and properties of Fano varieties.
result Proves existence of Kähler-Einstein metrics on Xm for m≥4 and on Ym for m=4,5. Study optimal degenerations of Fano threefolds, proving K-polystability and Kähler-Ricci solitons.
problem Optimal degenerations of K-unstable Fano threefolds.
method Explicitly determined degenerations, finding weighted K-polystable (X0,ξ0), studying moduli spaces. result One moduli space is isomorphic to the GIT-moduli space of biconic curves, the other is a single point.
The paper proves a CM line bundle is ample on K-moduli spaces.
problem Proving positivity of the CM line bundle on K-moduli spaces.
method Developed a new invariant for filtrations to test K-stability.
result Proves the CM line bundle is ample on reduced uniformly K-stable Fano varieties.
The paper proves the openness of K-semistability for Fano varieties.
problem Stability of K-semistability in families of log Fano pairs.
method By showing the stability threshold is a constructible function and proving special test configurations arise from log canonical places.
result The stability threshold is a constructible function on fibers, and any minimizer of the stability threshold exists.
Proves criteria for uniform K-stability of log Fano pairs.
problem Determining conditions for uniform K-stability of log Fano pairs.
method Analyzes criteria and proves equivalence to β-invariant having a positive lower bound.
result Uniform K-stability is equivalent to β-invariant having a positive lower bound.
Motivated by the study of Fano type varieties we define a new class of log pairs that we call asymptotically log Fano varieties and strongly asymptotically log Fano varieties. We study their properties in dimension two under an additional assumption of log smoothness, and give a complete classification of two dimension…
The paper proves the existence of Sasaki-Einstein metrics on specific Sasakian manifolds.
problem Proving the existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five.
method Deriving uniform L^{4}-bounds and analyzing the conic Sasaki-Ricci flow.
result Existence of conic Sasaki-Einstein metrics on log Fano Sasakian manifolds of dimension five.
Study shows Futaki invariant vanishes on most Fano threefolds.
problem Analyzing the Futaki invariant on K-polystable Fano threefolds.
method Examining the Futaki invariant as a map from the Kähler cone to the dual of the Lie algebra of the reduced automorphism group.
result Futaki invariant vanishes identically on most Fano threefolds.
We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.
Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.
problem Proper moduli spaces for K-polystable Q-Fano cones and their links.
method Algebraic construction using local normalized volume and higher Θ-stable reduction.
result Alternative algebraic proof of proper moduli spaces for Q-Fano varieties.
We study the Kahler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a Kahler-Einstein metric.
Counterexample disproves conjecture about Fano varieties with non-reductive automorphisms.
problem Disproving the conjecture about Loewy filtrations destabilizing non-reductive Fano varieties.
method Constructing a counterexample to the Loewy filtration conjecture.
result Found a Fano variety with non-reductive automorphism group that does not destabilize Loewy filtration.
We consider the Kähler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a Kähler-Einstein metric. The main ingredient is a result that says that a sufficiently small pert…
We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on…
We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…
New non-Kähler 3-folds constructed via log conifold transitions.
problem Constructing new non-Kähler 3-folds from Fano threefold pairs.
method Defining log conifold transitions and studying their deformation theory.
result Local smoothings of nodes can be lifted to global first-order deformations.
Solves modified conjecture for Fano manifolds using Ding stability.
problem Finding Kähler-Einstein metrics on Fano manifolds.
method Interprets Ding semistability and solves modified conjecture.
result Solves modified conjecture for coupled Kähler-Einstein metrics on Fano manifolds.
Study geometric properties of log Calabi-Yau manifolds, focusing on Fano manifolds with smooth or two proportional components.
problem Geometric properties of log Calabi-Yau manifolds in two specific cases.
method Analysis of various geometric properties, focusing on Bochner principle, local triviality, polystability, and compactifiability of universal cover.
result The universal cover of X∖D is a Calabi-Yau manifold of infinite topological type when D has two components. 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 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.
The study shows that normalized volumes of singularities can only jump down at countably many subvarieties.
problem Understanding the semi-continuity of normalized volumes in singularities.
method Using a flat family of klt singularities and an alternative characterization of K-semistability.
result K-semistability is a very generic or empty condition in Q-Gorenstein flat families of log Fano pairs.