arXiv research

Study finds limits for conical Kähler-Einstein metrics on unstable surfaces.

We show that the pair $(X, -K_X)$ is K-unstable for a del Pezzo manifold $X$ of degree five with dimension four or five. This disprove a conjecture of Odaka and Okada.

Segre varieties' hyperplane sections are unstable under certain conditions.

Study finds Kähler-Einstein metrics on two Pasquier varieties.

Proves properness of K-moduli spaces for Fano varieties.

The study examines K-polystability on Fano 4-folds with specific Lefschetz defects.

We prove that a pair (X, D) with X Fano and D a smooth anti-canonical divisor is K-unstable for negative angles, and K-semistable for zero angle.

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

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…

K-polystability of a polarised variety is an algebro-geometric notion conjecturally equivalent to the existence of a constant scalar curvature Kähler metric. When a variety is K-unstable, it is expected to admit a "most destabilising" degeneration. In this note we show that if such a degeneration exists, then the limit…

Equivalence proven between algebraic stability and geometric stability.

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

Tian's criterion for K-stability states that a Fano variety of dimension $n$ whose alpha invariant is greater than $\frac{n}{n+1}$ is K-stable. We show that this criterion is sharp by constructing singular Fano varieties with alpha invariants $\frac{n}{n+1}$ that are not K-polystable for sufficiently large $n$. We also…

On a K-unstable toric variety we show the existence of an optimal destabilising convex function. We show that if this is piecewise linear then it gives rise to a decomposition into semistable pieces analogous to the Harder-Narasimhan filtration of an unstable vector bundle. We also show that if the Calabi flow exists f…

We examine various examples of horosymmetric manifolds which exhibit interesting properties with respect to canonical metrics. In particular, we determine when the blow-up of a quadric along a linear subquadric admits Kähler-Einstein metrics, providing infinitely many examples of manifolds with no Kähler-Ricci solitons…

Classifies K-stable Fano varieties and finds new examples.

We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a we…

It is shown that any, possibly singular, Fano variety X admitting a Kahler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof exploits convexity properties of the Ding functiona…

The paper explores maximal destabilizers for both K-stability and Chow-stability in unstable situations.

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

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

New stability concept for Poisson structures leads to constant curvature metrics.

Introduces Poisson K-stability for Kähler manifolds and proves existence of constant scalar curvature structures.