Gábor Székelyhidi

University of Notre Dame
Department of Mathematics
277 Hurley
South Bend, IN 46556
email : gszekely at nd.edu

I am a Howard J. Kenna, C.S.C., Assistant Professor of Mathematics at the University of Notre Dame, interested in algebraic/differential geometry. My current research problem is understanding when extremal metrics exist on a polarized algebraic variety, and related questions.

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Teaching - Spring 2012

Topics in Analysis II - Extremal metrics
Problem set 1 - pdf - due February 20.
Problem set 2 - pdf - due March 30.
Problem set 3 - pdf - due May 8.

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Papers

• (with T. Collins) K-Semistability for irregular Sasakian manifolds
  [abstract] [pdf]

  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 results of Martelli-Sparks-Yau, and the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-Yau from this point of view.

• Filtrations and test-configurations
  [abstract] [pdf]

  We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of test-configurations, which arise naturally in several settings. We make some progress towards proving that if a manifold with no automorphisms admits a cscK metric, then it satisfies this stronger stability notion. Finally we discuss the relation with the birational transformations in the definition of b-stability.

• (with J. Song and B. Weinkove) The Kähler-Ricci flow on projective bundles
  [abstract] [pdf] to appear in Int. Math. Res. Not.

  We study the behaviour of the Kähler-Ricci flow on projective bundles. We show that if the initial metric is in a suitable Kähler class, then the fibers collapse in finite time and the metrics converge subsequentially in the Gromov-Hausdorff sense to a metric on the base.

• (with D. McFeron) On the positive mass theorem for manifolds with corners
  [abstract] [pdf] to appear in Comm. Math. Phys.

  We study the positive mass theorem for certain non-smooth metrics following P. Miao's work. Our approach is to smooth the metric using the Ricci flow. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.

• (with Renjie Feng) Periodic solutions of Abreu's equation
  [abstract] [pdf] to appear in Math. Res. Lett.

  We solve Abreu's equation with periodic right hand side, in any dimension. This can be interpreted as prescribing the scalar curvature of a torus invariant metric on an Abelian variety.

• On blowing up extremal Kähler manifolds
  [abstract] [pdf] to appear in Duke Math. J.

  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 Stoppa in this case. 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.

• (with J. Stoppa) Relative K-stability of extremal metrics
  [abstract] [pdf] J. Eur. Math. Soc. 13 (2011) n. 4, 899--909

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

• (with V. Tosatti) Regularity of weak solutions of a complex Monge-Ampère equation
  [abstract] [pdf] Analysis & PDE 4 (2011), n. 3, 369--378

  We prove the smoothness of weak solutions to an elliptic complex Monge-Ampère equation, using the smoothing property of the corresponding parabolic flow.

• (with O. Munteanu) On convergence of the Kähler-Ricci flow
  [abstract] [pdf] Comm. Anal. Geom. 19 (2011), n. 5, 887--904

  We study the convergence of the Kähler-Ricci flow on a Fano manifold under some stability conditions. More precisely we assume that the first eingenvalue of the $\bar\partial$-operator acting on vector fields is uniformly bounded along the flow, and in addition the Mabuchi energy decays at most logarithmically. We then give different situations in which the condition on the Mabuchi energy holds.

• Greatest lower bounds on the Ricci curvature of Fano manifolds
  [abstract] [pdf] Compositio Math. 147 (2011), 319--331

  On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric in c_1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin's continuity path for finding Kähler-Einstein metrics. We show that on P^2 blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

• The Kähler-Ricci flow and K-polystability
  [abstract] [pdf] Amer. J. Math. 132 (2010), 1077--1090

  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 perturbation of a cscK manifold admits a cscK metric if it is K-polystable.

• The Calabi functional on a ruled surface
  [abstract] [pdf] Ann. Sci. Éc. Norm. Supér. 42 (2009), 837--856

  We study the Calabi functional on a ruled surface over a genus two curve. For polarisations which do not admit an extremal metric we describe the behaviour of a minimising sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimising sequence.

• Optimal test-configurations for toric varieties
  [abstract] [pdf] J. Differential Geom. 80 (2008), 501--523

  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 for all time on a toric variety then it minimises the Calabi functional. 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.

• Extremal metrics and K-stability
  [abstract] [pdf] Bull. London Math. Soc. 39 (2007), 76--84

  We propose an algebraic geometric stability criterion for a polarised variety to admit an extremal Kähler metric. This generalises conjectures by Yau, Tian and Donaldson which relate to the case of Kähler-Einstein and constant scalar curvature metrics. We give a result in geometric invariant theory that motivates this conjecture, and an example computation that supports it.

• (with M. Laczkovich) Harmonic analysis on discrete Abelian groups
  [abstract] [link] Proc. Amer. Math. Soc. 133 (2005), 1581--1586
  Let G be an Abelian group and let C^G denote the linear space of all complex-valued functions defined on G equipped with the product topology. We prove that the following are equivalent. (i) Every nonzero translation invariant closed subspace of C^G contains an exponential; that is, a nonzero multiplicative function. (ii) The torsion free rank of G is less than the continuum.

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Thesis

The title of my PhD thesis is Extremal metrics and K-stability, supervised by Simon Donaldson.
[abstract] [pdf]

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Links

• Notre Dame Math Department
• Geometry at Imperial
• Columbia Math Department
• László Székelyhidi Jr.