arXiv research

Proves constant scalar curvature Kähler metrics are very general.

Proves results on K-stability using arcs and Mabuchi functional.

We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $β$-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stabilit…

Uniform K-stability ensures existence of special metrics on toric manifolds.

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant…

The paper simplifies K-stability conditions for spherical varieties.

Proves uniform K-stability is open in Kähler cone.

Article proves effective conditions for existence of Kähler metrics.

In this paper we prove that for toric varieties the uniform K-stability is the necessary condition for the existence of extremal metrics.

The paper generalizes K-stability results to singular and weighted settings.

We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties. To prove this result, we consider the behavior of the stability threshold in families. The stability threshold (also known as the delta-invariant) is a recently introduced invariant that is known to detect the K-s…

The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. …

The paper classifies test configurations and derives a criterion for uniform K-stability of certain algebraic varieties.

New stability criteria for Fano varieties using generalized b-divisors.

Equivalence proven between divisorial stability and quotient log divisorial stability.

Extremal metrics exist if uniformly $K$-stable over models.

In this paper, we discuss stable pairs, which were first studied by S. Paul, and give a proof for a result I learned from him. As a consequence, we will show that the K-stability implies the CM-stability.

From the work of Dervan-Keller, there exists a quantization of the critical equation for the J-flow. This leads to the notion of J-balanced metrics. We prove that the existence of J-balanced metrics has a purely algebro-geometric characterization in terms of Chow stability, complementing the result of Dervan-Keller. We…

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

Uniform K-stability of Calabi-Yau fibrations linked to base curve stability.

Decomposes J-energy into simpler intersection numbers for stability analysis.

Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as…

Uniform Closure Method and Bayes classifier perform similarly in classifying open knots.

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more gen…

We study a generalized Abreu Equation in $n$-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform $K$-stability.

We study the Abreu's equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.

We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.

Assume that a projective variety together with a polarization is uniformly K-stable. If the polarization is canonical or anti-canonical, then the projective variety is uniformly K-stable with respects to any polarization sufficiently close to the original polarization.

Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

An almost Kähler structure is {\it extremal} if the Hermitian scalar curvature is a Killing potential [29]. When the almost complex structure is integrable it coincides with extremal Kähler metric in the sense of Calabi [8]. We observe that the existence of an extremal {\it toric} almost Kähler structure of involutive …

Hensel-Przytycki-Webb proved that all curve graphs of orientable surfaces are 17-hyperbolic. In this paper, we show that curve graphs of non-orientable surfaces are 17-hyperbolic by applying Hensel-Przytycki-Webb's argument. We also show that arc graphs of non-orientable surfaces are 7-hyperbolic, and arc-curve graphs …

We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity should be reduced with respect to the group action. About this point we explain …

Let $X$ be any $\mathbb{Q}$-Fano variety and $\mathrm{Aut}(X)_0$ be the identity component of the automorphism group of $X$. Let $\mathbb{G}$ be a connected reductive subgroup of $\mathrm{Aut}(X)_0$ that contains a maximal torus of $\mathrm{Aut}(X)_0$. We prove that $X$ admits a Kähler-Einstein metric if and only if $X…

Uniform Ding stability implies existence of Kähler-Einstein metric on big anticanonical manifolds.

The purpose of this paper is to clarify all of the uniformly relatively Ding stable toric Fano threefolds and fourfolds as well as unstable ones. The key player in our classification result is the Mabuchi constants, which can be calculated by combinatorial data of the associated moment polytopes due to the work of Yao …

Derives formulas for Monge-Ampère measures and reduces complex conjectures to simpler existence problems.

We prove that any finite energy geodesic ray with a finite Mabuchi slope is maximal in the sense of Berman-Boucksom-Jonsson, and reduce the proof of the uniform Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler metrics to Boucksom-Jonsson's regularization conjecture about the convergence of non-Archime…

Bayesian structure learning is the NP-hard problem of discovering a Bayesian network that optimally represents a given set of training data. In this paper we study the computational worst-case complexity of exact Bayesian structure learning under graph theoretic restrictions on the super-structure. The super-structure …

Paper proves existence of unique constant scalar curvature Kähler metric under certain conditions.

Defines new stability conditions for Sasaki manifolds and extremal metrics.

New approach proves K-stability of Fano varieties.

Established a correspondence for toric fibrations using Delzant polytopes.

Effective Yau-Tian-Donaldson conjecture for spherical varieties.

We make some observation on the logarithmic version of K-stability.

Finite group action on K-stability results in standard stability.

Extends K-stability theory to projective klt pairs with a big anticanonical class.

Paper proposes a weak approximation of reflection coupling for non-convex optimization.