New methods for computing volumes and constructing Fano fibrations.
problem Understanding Fano fibrations and their weighted volumes.
method New methods for computing weighted volumes, including Laplace transforms and incomplete Gamma-functions. Conjectural construction of Fano fibrations.
result Conjectural construction of Fano fibrations with asymptotically conical bases from degenerating Fano fibrations.
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.
Introduces new stability concept for Fano fibrations.
problem Stability of Fano fibrations with singularities.
method Introduces f-stability and shows its implications. result Fibered semi log canonical singularities are restricted.
The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.
problem Understanding the relationship between Kähler-Ricci shrinkers and Fano fibrations.
method Using birational algebraic geometry, the paper proves properties of Kähler-Ricci shrinkers and formulates conjectures relating them to Fano fibrations.
result The existence of Kähler-Ricci shrinkers is conjectured to be related to K-stability of polarized Fano fibrations.
Inspired by the work of Gross on topological Mirror Symmetry we construct candidate Lagrangian torus fibration models for the 105 families of smooth Fano threefolds. We prove, in the case the second Betti number is one, that the total space of each fibration is homeomorphic to the expected Fano threefold, and show that…
Study Fano fibrations and Kähler-Einstein metrics on their bases.
problem Understand geometric structures and metrics on Fano fibrations.
method Construct (1,1)-forms and solve twisted Kähler-Einstein equations. result Singular Kähler metrics on Fano fibrations satisfy twisted Kähler-Einstein equations.
Study Fano fibrations and Kähler-Ricci flow singularities, proving diameter bounds and curvature estimates.
problem Analyzing Fano fibrations and Kähler-Ricci flow singularities.
method Developsing a singularity in finite time, using rational initial metrics and collapsing volume forms.
result Diameter bounds and curvature estimates for Fano fibrations and Kähler-Ricci flow singularities.
Simply-connected shrinking Kähler-Ricci solitons are proven.
problem Simply-connectedness of shrinking Kähler-Ricci solitons.
method Using results by Sun-Zhang and Wylie.
result Shrinking Kähler-Ricci solitons are simply-connected.
We study finite-time collapsing limits of the continuity method. When the continuity method starting from a rational initial Kähler metric on a projective manifold encounters a finite-time volume collapsing, this projective manifold admits a Fano fibration over a lower dimensional base. In this case, we prove the conti…
Homological mirror symmetry for toric Fano surfaces using Morse homotopy.
problem Establishing homological mirror symmetry for toric Fano surfaces.
method Applying SYZ construction and using Morse homotopy of the moment polytope.
result Homological mirror symmetry achieved for toric Fano surfaces.
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.
We show that if on a compact Kahler threefold there is a solution of the Kahler-Ricci flow which encounters a finite time collapsing singularity, then the manifold admits a Fano fibration. Furthermore, if there is finite time extinction then the manifold is Fano and the initial class is a positive multiple of the first…
In this paper, we show that along Q-Fano fibration, when general fibres, base and central fiber (with at worst Kawamata log terminal singularities)are K-poly stable then there exists a relative Kähler-Einstein metric. We introduce the fiberwise Kähler-Einstein foliation and we mention that the main difficulty…
Paper proves various types of varieties minimize a specific energy.
problem Minimizing a specific energy in various types of varieties.
method Analyzes various polarized varieties over Q. result Various types of varieties minimize the Arakelov K-energy.
Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.
problem Understanding the moduli spaces of quasimaps and Calabi-Yau fibrations.
method Constructing a projective K-moduli space of quasimaps and investigating relationships with Calabi-Yau fibrations.
result Entire quasi-projectivity and ampleness of the CM line bundle on the normalization of the K-moduli space of Calabi-Yau fibrations.
We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Grom…
We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Ka…
Extremal Kahler metrics and Sasaki-Einstein metrics characterized via coercive energy.
problem Characterizing extremal Kahler and Sasaki metrics using energy coercivity.
method Maximal complex torus, coercive weighted Mabuchi energy, K-polystability.
result Coercive weighted Mabuchi energy implies strict positivity of Donaldson-Futaki invariant and existence of extremal metrics.
Study of tangent bundle positivity on complex projective varieties.
problem Positivity of the second exterior power of tangent bundles on smooth complex projective varieties.
method Analyzes properties of tangent bundles and uses algebraic geometry techniques.
result Proves that up to a finite cover, the Albanese map is a locally trivial fibration with nef fibers.
We prove that any symplectic Fano 6-manifold M with a Hamiltonian S1-action is simply connected and satisfies c1c2(M)=24. This is done by showing that the fixed submanifold Mmin⊆M on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a 2-sphere or a po…
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.
Proves stability condition for Lagrangian sections in toric weak Fano manifolds.
problem Stability of Lagrangian sections in Calabi-Yau fibrations.
method SYZ transform, toric gamma theorem, Nakai-Moishezon criterion.
result Hamiltonian isotopy of Lagrangian sections under stability condition.
In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. A…
Kähler-Ricci flow shows type II singularity on Fano threefolds.
problem Understanding the behavior of Kähler-Ricci flow on Fano threefolds.
method Analyzing the Kähler-Ricci flow on Fano threefolds from a specific family.
result Kähler-Ricci flow develops type II singularity on Fano threefolds from the specified family.
Uniqueness proven for a specific type of complex manifold's solitons.
problem Proving uniqueness of asymptotically conical shrinking gradient Kähler-Ricci solitons.
method Using a method to show uniqueness of the soliton vector field, which can be applied more widely.
result A noncompact complex manifold admits only one such soliton.
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.
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.
New toric Fano manifolds found without extremal Kähler metrics.
problem Finding toric Fano manifolds without extremal Kähler metrics.
method Constructing specific toric Fano manifolds of dimensions 10 and n (n≥11) that do not admit extremal Kähler metrics.
result Existence of toric Fano manifolds of dimension 10 and higher that do not admit extremal Kähler metrics.
Paper studies limits of Kähler-Ricci flow on Fano G-manifolds.
problem Analyzing the limits of Kähler-Ricci flow on Fano G-manifolds.
method Proves the Gromov-Hausdorff limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
result The limit of Kähler-Ricci flow on Fano G-manifolds is a horosymmetric variety.
The study classifies complex smooth Fano varieties with large pseudoindex.
problem Classifying Fano varieties with specific properties.
method Analyzing Fano varieties with large pseudoindex and Picard number greater than one.
result Classification of Fano varieties with pseudoindex at least n-2 and Picard number greater than one.
New approach proves K-stability of Fano varieties.
problem Proving K-stability of Fano varieties.
method Developed a general approach using admissible flags.
result Proved K-stability of smooth Fano hypersurfaces of index two.
K-stability proven for a specific type of Fano threefold.
problem Proving K-stability of Fano threefolds.
method Analyzing double covers of blow-ups with specific branch divisors.
result Proven K-stability of Fano threefolds of rank 2 and degree 14.
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.
We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension a…
Study calculates volumes of Fano K-moduli spaces in various dimensions.
problem Computing volumes of Fano K-moduli spaces in different dimensions.
method Computed volumes of Fano K-moduli spaces of Quartic del Pezzo and log Fano hyperplane arrangements in dimensions one and two.
result Relates computed volumes to Weil-Petersson volumes, extending the Weil-Petersson metric to the log case.
Classifies Fano varieties with large pseudoindex and non-free rational curves.
problem Classifying Fano varieties with specific properties.
method Extremal contractions and classification of varieties.
result Complete classification of Fano n-folds with pseudoindex at least n−2 and Picard number greater than one. 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.
We prove the Yau-Tian-Donaldson's conjecture for any Q-Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the pre…
Study positive characteristic Fano 4-folds with nef tangent bundles.
problem Positive characteristic version of the Campana-Peternell conjecture for Fano 4-folds.
method Analyzes Fano 4-folds with nef tangent bundles in positive characteristic.
result Affirmative answer for Fano 4-folds with Picard number > 1 and nef tangent bundle.
The paper proves Mabuchi solitons and constants on Fano admissible manifolds.
problem Existence of Mabuchi solitons on Fano admissible manifolds.
method Defined Mabuchi solitons and constants, proved existence and non-existence.
result Fano admissible manifolds admit Mabuchi solitons if and only if the Mabuchi constant is less than 1.
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.
We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular att…
New research confirms Kähler-Einstein metrics for all Fano threefolds of degree 22.
problem Identifying Fano threefolds of degree 22 with Kähler-Einstein metrics.
method Analyzing the structure and properties of Fano threefolds of Picard rank one with anti-canonical degree 22.
result All remaining Fano threefolds of degree 22 admit Kähler-Einstein metrics.
We exhibit the first non-trivial concrete examples of Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifolds in all complex dimensions bigger than two (Fano K-moduli spaces). We also discuss potential applications to explicit study of moduli spaces of K-stable Fano manifolds with large an…
Sharp bounds on Fano varieties' heights proven for specific cases.
problem Determining the maximal height of arithmetic Fano varieties.
method Logarithmic extension of a conjecture, applied to specific Fano varieties.
result The conjecture settled for specific cases, including hypersurfaces and toric varieties.
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. The study classifies Fano varieties with specific pseudoindex.
problem Classifying Fano varieties with a particular pseudoindex.
method Classification based on birational contractions and pseudoindex condition.
result Classification of Fano varieties with pseudoindex equal to half of their dimension.
In this paper, we prove an existence result for Kähler-Einstein metrics on Q-Fano compactifications of Lie groups. As an application, we classify Q-Fano compactifications of SO4(C) which admit a Kähler-Einstein metric with the same volume as that of a smooth Fano compactification of $SO_4…