arXiv research

Study calculates volumes of Fano K-moduli spaces in various dimensions.

Short note shows unbounded dimensions in Fano K-moduli spaces.

The paper proves a CM line bundle is ample on K-moduli spaces.

Proves properness of K-moduli spaces for Fano varieties.

Study of K-moduli of prime Fano threefolds of genus twelve, proving boundary purely divisorial.

Study shows K-moduli spaces of curves on quadrics and K3 surfaces match with VGIT quotients.

We construct proper good moduli spaces parametrizing K-polystable $\mathbb{Q}$-Gorenstein smoothable log Fano pairs $(X, cD)$, where $X$ is a Fano variety and $D$ is a rational multiple of the anti-canonical divisor. We then establish a wall-crossing framework of these K-moduli spaces as $c$ varies. The main applicatio…

Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.

Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.

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…

Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.

Cubic fourfolds have K-stability and admit Kähler-Einstein metrics.

Proves finitely generated associated graded rings for valuations on log Fano pairs.

The paper proves K-stability of special Gushel-Mukai manifolds.

Optimizes bounds for threefold singularity volumes.

A smooth curve found in a space of special surfaces.

Log minimality proven for weak K-moduli compactifications of Calabi-Yau varieties.

Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.

Develops moduli theory for Calabi-Yau pairs, constructing a projective space.

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…

We construct a geometrically compactified moduli algebraic space of Kahler-Einstein Fano manifolds.

Constructs projective moduli spaces for Calabi-Yau pairs.

Defines a new metric on Fano Kaehler-Ricci solitons.

We investigate Fano schemes of conditionally generic intersections, i.e. of hypersurfaces in projective space chosen generically up to additional conditions. Via a correspondence between generic properties of algebraic varieties and events in probability spaces that occur with probability one, we use the obtained resul…

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit …

We prove that Kahler-Einstein Fano manifolds with finite automorphism groups form Hausdorff moduli algebraic space with only quotient singularities. We also discuss the limits as Q-Fano varieties which should be put on the boundary of its canonical compactification.

We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (resp. no smaller than) $\frac{1}{2}$ is K-stable (resp. K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension $n$ is at least $\frac{1}{n+1}$ (under mild assumptions) an…

The paper proves Mabuchi solitons and constants on Fano admissible manifolds.

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $Θ$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove…

Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.

Study optimal degenerations of Fano threefolds, proving K-polystability and Kähler-Ricci solitons.

Study lower bounds on modified K-energy on Fano manifolds with Kähler-Ricci solitons.

The study finds Kähler-Einstein metrics on certain Fano varieties of type AIII.

The paper classifies sextic curves on a Fano 3-fold with rational Galois covers in 3D space.

Sharp bounds on Fano varieties' heights proven for specific cases.

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…

We study the behavior under Gromov-Hausdorff convergence of the spectrum of weighted $\barpartial$-Laplacian on compact Kähler manifolds. This situation typically occurs for a sequence of Fano manifolds with anticanonical Kähler class. We apply it to show that, if an almost smooth Fano-Ricci limit space admits a Kähler…

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kähler-Ricci solitons. This naturally enlarges the moduli space of Fano manifolds with Kähler-Einstein metrics, which was constructed by Odaka and Li-Wang-Xu. We discover a moment map picture for Kähler-Ricci solitons, and give comp…

We consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent sub…

We survey recent results on the existence of Kähler-Einstein metrics on certain smoothable Fano varieties, focusing on the importance of such metrics in the construction of compact algebraic moduli spaces of K-polystable Fano varieties. Moreover, we give some applications and we discuss some natural problems which dese…

Study moduli space of cscK surfaces around toric ones, introducing foldable surfaces.

In this short paper, we improve the result of Phong-Song-Sturm on degeneration of Fano Kähler-Ricci solitons by removing the assumption on the uniform bound of the Futaki invariant. Let $\mathcal{KR}(n)$ be the space of Kähler-Ricci solitons on $n$-dimensional Fano manifolds. We show that after passing to a subsequence…

New proof for curvature and diameter estimates on Fano manifolds.

It is known that a necessary condition for the existence of Kähler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nyström, it was generalized for (singular) Fano varieties and the notion of algebro-geometric stability of the pair $(M,V)$ of a Fano man…

In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in the direction of the study of moduli spaces and their compactifications.

Sharp bounds on K-semistable Fano varieties for low dimensions.

In this note, we prove that there is a canonical continuous Hermitian metric on the CM line bundle over the proper moduli space $\bar{\mathcal{M}}$ of smoothable Kahler-Einstein Fano varieties. The curvature of this metric is the Weil-Petersson current, which exists as a positive (1,1)-current on $\bar{\mathcal{M}}$ an…

The study examines conical Kähler-Einstein metrics on specific Fano manifolds and their behavior at the limit angle.