Short note shows unbounded dimensions in Fano K-moduli spaces.
problem Unboundedness of Fano K-moduli space dimensions.
method Analyzes Fano varieties and their K-moduli spaces.
result Dimension of K-moduli space can be unbounded.
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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.
Log minimality proven for weak K-moduli compactifications of Calabi-Yau varieties.
problem Constructing weak K-moduli compactifications of Calabi-Yau varieties.
method Revisiting classical problem, proving log minimality of normalizations under conditions.
result Log minimality of weak K-moduli compactifications under certain conditions.
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.
A smooth curve found in a space of special surfaces.
problem Understanding the structure of K-polystable del Pezzo surfaces.
method Explicit construction of a component in the K-moduli space.
result Found a smooth rational curve in the K-moduli space.
Study shows K-moduli spaces of curves on quadrics and K3 surfaces match with VGIT quotients.
problem Understanding K-moduli spaces of curves on quadrics and K3 surfaces.
method Using log Fano pairs and VGIT quotients, the study compares K-moduli spaces of curves on and quartic hyperelliptic K3 surfaces.
result K-moduli spaces of curves on quadrics and K3 surfaces form a natural interpolation.
Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.
problem Understanding the boundary of K-moduli of prime Fano threefolds of genus twelve.
method Developed a modular relation between Fano threefolds and their anticanonical K3 surfaces, proving forgetful morphism is an open immersion.
result Proved the boundary of K-moduli of is purely divisorial and consists of four irreducible components.
We construct proper good moduli spaces parametrizing K-polystable -Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as varies. The main applicatio…
Semi-toric and toroidal compactifications as log minimal models, and applications to weak K-modulimath.AG
Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
problem Characterizing and applying toroidal and semi-toric compactifications to weak K-moduli.
method Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.
result Different proof of a theorem of Alexeev-Engel on weak K-moduli compactifications.
Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.
problem Understanding the moduli spaces of quartic K3 surfaces and their birational models.
method Interpolates between GIT and Baily-Borel moduli spaces, describes wall crossings, and classifies degenerations.
result Verifies Laza-O'Grady's prediction and classifies Gorenstein canonical Fano degenerations of \(\mathbb{P}^3\).
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.
In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that …
Develops moduli theory for Calabi-Yau pairs, constructing a projective space.
problem Constructing a moduli space for Calabi-Yau pairs at the Calabi-Yau wall.
method Develops moduli theory, proving S-completeness and -reductivity, constructing projective moduli space.
result Constructs a projective moduli space for degenerate pairs.
The paper proves K-stability of special Gushel-Mukai manifolds.
problem Proving K-stability of special Gushel-Mukai manifolds.
method Analyzing the structure of Gushel-Mukai manifolds and their K-stability.
result General special Gushel-Mukai n-folds are K-stable for 3 ≤ n ≤ 6.
Explicit Gromov-Hausdorff compactifications of moduli spaces of Kähler-Einstein Fano manifoldsmath.DG
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…
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.
Cubic fourfolds have K-stability and admit Kähler-Einstein metrics.
problem Understanding K-stability and Kähler-Einstein metrics for cubic fourfolds.
method Local volume estimates and Ambro-Kawamata's non-vanishing theorem.
result All smooth cubic fourfolds admit Kähler-Einstein metrics.
Constructs projective moduli spaces for Calabi-Yau pairs.
problem Creating projective moduli spaces for Calabi-Yau surface pairs.
method Constructs projective asymptotically good moduli spaces.
result Provides a wall crossing between KSBA and K-moduli spaces.
We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric…
Study moduli space of cscK surfaces around toric ones, introducing foldable surfaces.
problem Moduli space of cscK surfaces around toric ones.
method Introduced foldable surfaces and classified them. Studied moduli space locally.
result Moduli space locally modeled on a finite quotient of a toric affine variety with terminal singularities.
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.
Finite generation for valuations computing stability thresholds and applications to K-stabilitymath.AG
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.
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.