arXiv research

Equivalence proven between divisorial stability and quotient log divisorial stability.

We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the …

We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability threshold. When such a divisorial valuation exists, we show that the Fano variety degenerates to a uniquely determined twisted K-polystable Fan…

We prove that among all Kollár components obtained by plt blow ups of a klt singularity $o \in (X, D)$, there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Kollár component exists, it uniquely minimizes the normalized volume function introduced in [Li15a] among all divisorial valu…

Proves existence of Kähler-Einstein metrics in big cohomology classes.

Proves properness of K-moduli spaces for Fano varieties.

For any $\mathbb{Q}$-Gorenstein klt singularity $(X,o)$, we introduce a normalized volume function $\widehat{\rm vol}$ that is defined on the space of real valuations centered at $o$ and consider the problem of minimizing $\widehat{\rm vol}$. We prove that the normalized volume has a uniform positive lower bound by pro…

Kähler-Ricci flow smooths out positive closed currents with divisorial singularities

Introduces new hyperbolicity concepts for complex manifolds.

New hyperbolicity concepts expand manifold study.

We study the behaviour of Chern numbers of three dimensional terminal varieties under divisorial contractions.

We discuss a technique to construct Ricci-flat hermitian metrics on complements of (some) anticanonical divisors of almost homogeneous manifolds and discuss when this metric is complete and Kähler. This construction has a strong interplay with invariance groups of the same dimension as the manifold acting with an open …

Paper rigorously defines Feynman graph integrals on Kähler manifolds.

We compute a closed formula for the class of the closure of the locus of curves in $\overline{\mathcal{M}}_g$ that admit an abelian differential of signature $κ=(k_1,...,k_{g-2})$.

We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimens…

We consider two applications of the strata of differentials of the second kind (all residues equal to zero) with fixed multiplicities of zeros and poles: Positivity: In genus $g=0$ we show any associated divisorial projection to $\overline{\mathcal{M}}_{0,n}$ is $F$-nef and hence conjectured to be nef. We compute the c…

We prove the existence and uniqueness of the weak Kahler-Ricci flow on projective varieties with log terminal singularities. It is also shown that the weak Kahler-Ricci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an analytic version of the Minimal Model Progra…

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

We compute the hybrid limit (in the sense of Boucksom-Jonsson) of the family of Kähler-Einstein volume forms on a degeneration of canonically polarized manifolds. The limit measure is a weighted sum of Dirac masses at divisorial valuations, determined by the natural algebro-geometric limit of the family. We also make s…

Study shows hypercomplex twistor spaces lack divisors and special metrics.

We establish the existence of the K"ahler-Ricci flow on projective varieties with log canonical singularities. This generalizes some of the existence results of Song-Tian \cite{ST3} in case of projective varieties with klt singularities. We also prove that the normalized K"ahler-Ricci flow will converge to the \ka-Eins…

Study shows Kähler-Einstein metric singularities linked to curvature.

In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divis…

Study collapsing Calabi-Yau metrics and flows on fiber spaces.

The paper defines two types of hyperbolicity for complex manifolds and proves related results.

The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.

This is a continuation to the paper [arXiv:1511.08164] in which a problem of minimizing normalized volumes over $\mathbb{Q}$-Gorenstein klt singularities was proposed. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties. In particul…

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a ration…

Let $V$ be a regular neighborhood of a negative chain of $2$-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let $B_{p,q}$ be a rational homology ball which is smoothly embedded in $V$. Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequen…

New method constructs small symplectic 4-manifolds via contact gluing.

Introduces stability conditions for polarized varieties, linking to K-stability.

Introduces valuative stability for polarised varieties, equivalent to K-stability.

The homology groups of many natural sequences of groups $\{G_n\}_{n=1}^{\infty}$ (e.g. general linear groups, mapping class groups, etc.) stabilize as $n \rightarrow \infty$. Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequ…

We can talk about two kinds of stability of the Ricci flow at Ricci flat metrics. One of them is a linear stability, defined with respect to Perelman's functional $\mathcal{F}$. The other one is a dynamical stability and it refers to a convergence of a Ricci flow starting at any metric in a neighbourhood of a considere…

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

In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow. We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.

Study on stability of hyperkähler flow in 4-manifolds.

Study max- and min-stability under first-order stochastic dominance, finding new functional characterizations.

For a polarized algebraic manifold $(X,L)$, let $T$ be an algebraic torus in the group of all holomorphic automorphisms of $X$. Then strong relative K-stability will be shown to imply asymptotic relative Chow-stability. In particular, by taking $T$ to be trivial, we see that asymptotic Chow-stability follows from stron…

Introduces new stability concept for Fano fibrations.

The paper proves stability for contact groupoids and deformations.

Solves modified conjecture for Fano manifolds using Ding stability.

This paper enhances stability selection by evaluating overall results robustness and identifying optimal regularization values.

This work explores the trade-offs between stability and accuracy in statistical estimation.

In this note, we shall show that the Chow-stability and the Hilbert-stability in GIT asymptotically coincide.

The paper examines stability of harmonic and symphonic maps with forms and potentials.

We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear stability of such gradient Ricci solitons.

New stability measures for similar features improve feature selection accuracy.