### GABOR SZEKELYHIDI THESIS

We prove the conjecture stated in [32], and we relate this result to the K-stability of blown up manifolds. Extremal metrics and K-stability [ abstract ] [ pdf ] Bull. Lejmi The J-flow and stability [ abstract ] [ pdf ] Advances in Math. Weinkove Gauduchon metrics with prescribed volume form [ abstract ] [ pdf ] Acta Math. 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. As an application we show that the five-sphere admits infinitely many families of Sasaki-Einstein metrics.

We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. McFeron On the positive mass theorem for manifolds with corners [ abstract ] [ pdf ] Comm. The Calabi functional on a ruled surface [ abstract ] [ pdf ] Ann. As an application we show that the blowup of a Kahler-Einstein manifold at a point admits a constant scalar curvature Kahler metric in classes that make the exceptional divisor small, if it is K-polystable with respect to these classes. A, 36 no. Collins Sasaki-Einstein metrics and K-stability [ abstract ] [ pdf ] to appear in Geom. The method also applies to analogous equations on compact Riemannian manifolds.

Our main result is that each tangent cone is homeomorphic to a normal affine variety. Finally we prove a general existence result on complex tori.

## Mathematics Genealogy Project

Stoppa Relative K-stability of extremal metrics [ abstract ] [ pdf ] J. Greatest lower bounds on the Ricci curvature of Fano manifolds [ abstract ] [ pdf ] Compositio Math. We show that on a Kahler manifold whether the J-flow converges or not is independent of the chosen background metric in its Kahler class. Our approach is to smooth the metric using the Ricci flow. Extremal thdsis and K-stability [ abstract ] [ pdf ] Bull.

We first discuss the Yau-Tian-Donaldson conjecture relating the existence of extremal metrics to an algebro-geometric stability notion and we give some example settings where this conjecture szekeoyhidi been established. Our approach is to use a linear expression of the eigenvalues of the Hessian instead of quotients of elementary symmetric functions. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from Finally we discuss the relation with the birational transformations in the definition of b-stability.

In this short note we prove that if the curvature tensor is uniformly bounded along the Calabi flow and the Mabuchi energy is proper, then the flow converges to a constant scalar curvature metric. Collins Sasaki-Einstein metrics and K-stability [ abstract ] [ pdf ] to appear in Geom.

# Gabor Szekelyhidi homepage

This can be interpreted as prescribing the scalar curvature of a torus invariant metric on an Abelian variety. Weinkove Gauduchon metrics with prescribed volume form [ abstract ] [ pdf ] Acta Math. We study the J-flow from the point of view of an algebro-geometric stability gabo. We study the positive mass theorem for certain non-smooth metrics following P. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces.

## Gábor Székelyhidi

We show that the blowup of an extremal Kahler manifold at a relatively stable point in the sense of GIT fhesis an extremal metric in Kahler classes that make the exceptional ganor sufficiently small, extending a result of Arezzo-Pacard-Singer. Seyyedali Extremal metrics on blowups along submanifolds [ abstract ] [ pdf ] to appear in J. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising sequence.

Lecture Schedule ; Information on projects.

This extends to the irregular thseis the orbifold K-semistability of Ross-Thomas. As an application we show that the five-sphere admits infinitely many families of Sasaki-Einstein metrics.

We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. We also study the K-polystability of these blowups, sharpening a result of Stoppa in this case.

# [math/] Extremal metrics and K-stability (PhD thesis)

Optimal test-configurations for toric zsekelyhidi [ abstract ] [ pdf ] J. Collins Convergence of the J-flow on toric manifolds [ abstract ] [ pdf ] J. On a K-unstable toric variety we show the existence of an optimal destabilising convex function. In this case the infimum of the Calabi functional is given by the supremum of the normalised Futaki invariants over all destabilising test-configurations, as predicted by a conjecture of Donaldson.

We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it. Fully non-linear elliptic gablr on compact Hermitian manifolds [ abstract ] [ pdf ] J. We extend their approach to the setting where only a lower bound for the Ricci curvature is assumed.

In particular we show that the maximal possible cone angle is in general smaller than the invariant R M. As an application we show that the blowup of a Kahler-Einstein manifold at a point admits a constant scalar curvature Kahler metric in classes that make the exceptional divisor small, if it is K-polystable with respect to these classes.

We study this discrepancy from the point of view of log K-stability.