arXiv research

This paper surveys geometric invariant theory for Yang-Mills equations on Riemann surfaces.

Study on semistable points and convexity of gradient maps for group actions.

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 …

New polystability theory connects Calabi-Yau varieties to gravitational instantons.

The paper studies semistability in polarized toric manifolds and their divisors.

The paper defines and proves conditions for numerical semistability of smooth toric varieties.

Introduces semistability in geometric group theory and provides techniques to prove it.

Groups with semistable peripheral subgroups are semistable.

Characterizes K-semistability for log Fano cone singularities.

Study actions of mapping class groups on surface representations, proving finite image for certain representations.

Alternative proof of semipositivity and nefness for K-semistable log-Fano pairs.

Study shows volume limit for K-semistable Fano manifolds.

We introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case the orbifold K-semistability of Ross-Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, we show how one can recover the volume minimization …

New finding on K-semistability in optimal degenerations.

We study the basic properties of Higgs sheaves over compact Kähler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of a…

No semistability found for Calabi-Yau metrics near cones.

The paper proves the openness of K-semistability for Fano varieties.

Reductive automorphism groups for K-polystable Fano pairs proved.

New findings link K-semistability to volume minimization for Fano varieties.

Proves stable degeneration preserves symplectic forms and confirms Kaledin's conjecture.

We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is se…

Let $Δ\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_Δ$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_Δ$ on $X_Δ$ associated with $Δ$. In the present paper, we give a necessary and sufficient …

Continuity shown for Yang-Mills flow on semistable bundles.

The paper proves hyperbolic groups are semistable and their boundaries are linearly connected.

We initiate the study of the asymptotic topology of groups that can be realized as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the $H_1$-semistability conjecture of Geoghegan for holomorphically…

Researchers prove K-semistability for cscK manifolds with transcendental cohomology.

In this note, using the recent compactness results of Tian and Chen-Donaldson-Sun, we prove the K-semistable version of Yau-Tian-Donaldson correspondence for Fano manifolds.

Odaka and Wang proved the intersection formula for the Donaldson-Futaki invariant. In this paper, we generalize this result for the higher Futaki invariants which are obstructions to asymptotic Chow semistability.

The paper proves a unique Kollár component is K-semistable among plt blow ups of a klt singularity.

Let $G$ be a connected reductive affine algebraic group defined over $\mathbb C$, and let $Γ$ be a cocompact lattice in $G$. We prove that any invariant bundle on $G/Γ$ is semistable.

We study the existence of canonical Kähler metrics on the projectivisation of strictly Mumford semistable holomorphic vector bundles over a complex curve. We also provide an algebro-geometric characterization of these metrics.

Study proves Higgs fields non-existent on Calabi-Yau manifolds.

The paper classifies and computes limits of equivariant compactifications of groups.

A valuation minimizes volume for K-semistable singularities.

Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle …

Study homotopy groups in GIT quotients using transversality methods.

Let $H$ be a semisimple algebraic group. We prove the semistable reduction theorem for $μ$--semistable principal $H$--bundles over a {\it smooth projective variety $X$} defined over the field $\bc$. When $X$ is a {\it smooth projective surface} and $H$ is simple, we construct the algebro--geometric Donaldson--Uhlenbeck…

We review the notions of (weak) Hermitian-Yang-Mills structure and approximate Hermitian-Yang-Mills structure for Higgs bundles. Then, we construct the Donaldson functional for Higgs bundles over compact Kähler manifolds and we present some basic properties of it. In particular, we show that its gradient flow can be wr…

Characterizes Q-Gorenstein singularities via K-stability.

In this note, by using the Yang-Mills-Higgs flow, we show that semistable Higgs bundles with vanishing the first and second Chern numbers over compact Käher manifolds must admit a filtration whose quotients are Hermitian flat Higgs bundles.

We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact Kähler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they…

Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.

Study confirms boundedness of certain singularities in log Fano geometry.

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

The notion of Berman-Gibbs stability was originally introduced by Robert Berman for $\mathbb{Q}$-Fano varieties $X$. We show that the pair $(X, -K_X)$ is K-stable (resp. K-semistable) provided that $X$ is Berman-Gibbs stable (resp. semistable).

The normalized volume is lower semicontinuous in klt singularities.

We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus $g$, the number $N$ of non-separating vanishing cycles and the number $D$ of singular fibers satisfy the inequality $N \…

The study shows that normalized volumes of singularities can only jump down at countably many subvarieties.