arXiv research

Unique K-polystable degenerations for Fano varieties confirmed.

Proves unique degeneration of log Fano fibration germs.

Stability of weighted extremal manifolds proven through blowups.

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension a…

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

We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to…

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…

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…

Analytic K-semistability connects curvature to metric existence.

Proves properness of K-moduli spaces for Fano varieties.

In this paper we study K-polystability of arbitrary (possibly non-projective) compact Kähler manifolds admitting holomorphic vector fields. As a main result, we show that existence of a constant scalar curvature Kähler (cscK) metric implies 'geodesic K-polystability', in a sense that is expected to be equivalent to K-p…

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

In this follow up work to [45, 33, 32, 46] we introduce and study a notion of geodesic stability restricted to rays with prescribed singularity types. A number of notions of interest fit into this framework, in particular algebraic- and transcendental K-polystability, equivariant K-polystability, and the geodesic K-pol…

Proves K-polystability for Kähler-Ricci shrinkers with decaying curvature.

A smooth curve found in a space of special surfaces.

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

Let $P(E)$ be the projectivization of a holomorphic vector bundle $E$ over a compact complex curve $C$. We characterize the existence of an extremal Kähler metric on the ruled manifold $P(E)$ in terms of relative K-polystability and the fact that $E$ decomposes as a direct sum of stable bundles.

It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant…

Equivalence proven between algebraic stability and geometric stability.

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…

The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.

We consider the Kähler-Ricci flow on a Fano manifold. We show that if the curvature remains uniformly bounded along the flow, the Mabuchi energy is bounded below, and the manifold is K-polystable, then the manifold admits a Kähler-Einstein metric. The main ingredient is a result that says that a sufficiently small pert…

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…

Survey on Kähler-Einstein and weighted solitons on Fano manifolds.

The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs.…

We provide a moment map interpretation for the coupled Kähler-Einstein equations introduced by Hultgren and Witt Nyström, and in the process introduce a more general system of equations, which we call coupled cscK equations. A differentio-geometric formulation of the corresponding Futaki invariant is obtained and a not…

In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that …

We show that the blowup of an extremal Kahler manifold at a relatively stable point in the sense of GIT admits an extremal metric in Kahler classes that make the exceptional divisor sufficiently small, extending a result of Arezzo-Pacard-Singer. We also study the K-polystability of these blowups, sharpening a result of…

K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical Kähler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical not…

We show that if a polarised manifold admits an extremal metric then it is K-polystable relative to a maximal torus of automorphisms.

In this article, we completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

Constructs optimal symplectic connections for Kaehler metrics on holomorphic submersions.

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

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

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

We prove the existence of Kahler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties, and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings.

We study the Kahler-Ricci flow on Fano manifolds. We show that if the curvature is bounded along the flow and if the manifold is K-polystable and asymptotically Chow semistable, then the flow converges exponentially fast to a Kahler-Einstein metric.

Study shows Futaki invariant vanishes on most Fano threefolds.

Parabolic structures with rational weights encode certain iterated blowups of geometrically ruled surfaces. In this paper, we show that the three notions of parabolic polystability, K-polystability and existence of constant scalar curvature Kähler metrics on the iterated blowup are equivalent, for certain polarizations…

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…

It is known that the automorphism group of a K-polystable Fano manifold is reductive. Codogni and Dervan construct a canonical filtration of the section ring, called Loewy filtration, and conjecture that the Loewy filtration destabilizes any Fano variety with non-reductive automorphism group. In this note, we give a co…

We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existe…

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…

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

We prove the Yau-Tian-Donaldson's conjecture for any $\mathbb{Q}$-Fano variety that has a log smooth resolution of singularities such that the discrepancies of all exceptional divisors are non-positive. In other words, if such a Fano variety is K-polystable, then it admits a Kähler-Einstein metric. This extends the pre…

Study links K-stability of certain surfaces to binary forms, proving stability and non-stability conditions.