Gábor Székelyhidi
University of Notre Dame
Department
of Mathematics
277 Hurley
South Bend, IN 46556
email : gszekely at nd.edu
I am an Associate 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.
CV
Teaching
Fall 2016
Basic Complex Analysis I,
office hours on Wednesdays 910:30am, or by appointment.
Homework
Problem sets:
1
Previous
teaching
Papers

(with T. Collins)
SasakiEinstein metrics and Kstability
[abstract] [pdf]
We show that a polarized affine variety admits a Ricci flat Kähler cone metric, if it is Kstable. This generalizes ChenDonaldsonSun's solution of the YauTianDonaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the fivesphere admits infinitely many families of SasakiEinstein metrics.

(with B. Weinkove)
On a constant rank theorem for nonlinear elliptic PDEs
[abstract] [pdf]
to appear in Dicrete Contin. Dyn. Syst. Ser. B
We give a new proof of BianGuan's constant rank theorem
for nonlinear elliptic equations. Our approach is to use a linear expression
of the eigenvalues of the Hessian instead of quotients of elementary
symmetric functions.

(with V. Datar)
KählerEinstein metrics along the smooth continuity
method
[abstract] [pdf]
to appear in GAFA
We show that if a Fano manifold M is Kstable with respect to special
degenerations equivariant under a compact group of automorphisms, then M
admits a KählerEinstein metric. This is a strengthening of the solution of
the YauTianDonaldson conjecture for Fano manifolds by ChenDonaldsonSun, and
can be used to obtain new examples of KählerEinstein manifolds. We also give
analogous results for twisted KählerEinstein metrics and KählerRicci
solitons.

(with V. Tosatti, B. Weinkove)
Gauduchon metrics with prescribed volume form
[abstract] [pdf]
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the ChernRicci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.

Fully nonlinear elliptic equations on compact Hermitian manifolds
[abstract] [pdf]
to appear in J. Differential Geom.
We derive a priori estimates for solutions of a general class of fully nonlinear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex MongeAmpère, Hessian and inverse Hessian equations. As an application we solve a class of Hessian quotient equations on Kähler manifolds assuming the existence of a suitable subsolution. The method also applies to analogous equations on compact Riemannian manifolds.

(with T. Collins)
Convergence of the Jflow on toric manifolds
[abstract] [pdf]
to appear in J. Differential Geom.
We show that on a Kahler manifold whether the Jflow converges or not is independent of the chosen background metric in its Kahler class. On toric manifolds we give a numerical characterization of when the Jflow converges, verifying a conjecture of Lejmi and the second author in this case. We also strengthen existing results on more general inverse sigma_k equations on Kahler manifolds.

Extremal Kähler metrics
[abstract] [pdf]
Proceedings of the ICM, 2014
This paper is a survey of some recent progress on the study of
Calabi's extremal Kähler metrics.
We first discuss the YauTianDonaldson conjecture relating the
existence of extremal metrics to an algebrogeometric stability
notion and we give some example settings where this conjecture has
been established. We then turn to the question of what one expects
when no extremal metric exists.

The partial C^{0}estimate along the continuity
method
[abstract] [pdf]
to appear in J. Amer. Math. Soc.
We prove that the partial C^{0}estimate holds for metrics along Aubin's continuity method for finding KählerEinstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of ChenDonaldsonSun on the analogous problem for conical KählerEinstein metrics.

(with M. Lejmi)
The Jflow and stability
[abstract] [pdf]
to appear in Advances in Math.
We study the Jflow from the point of view of an algebrogeometric stability condition. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by FangLai is reflected by the optimal destabilizer. Finally we prove a general existence result on complex tori.

Blowing up extremal Kähler manifolds II
[abstract] [pdf]
to appear in Invent. Math.
This is a continuation of the work of ArezzoPacardSinger and the author on
blowups of extremal Kähler manifolds. We prove the conjecture stated in [32],
and we relate this result to the Kstability of blown up manifolds. As an
application we prove that if a Kähler manifold M of dimension greater than 2
admits a cscK metric, then the blowup of M at a point admits a cscK metric if
and only if it is Kstable, as long as the exceptional divisor is sufficiently
small.

A remark on conical KählerEinstein metrics
[abstract] [pdf]
Math. Res. Lett. 20 (2013) n. 3., 581590
We give some nonexistence results for KählerEinstein metrics with conical singularities along a divisor on Fano manifolds. In particular we show that the maximal possible cone angle is in general smaller than the invariant R(M). We study this discrepancy from the point of view of log Kstability.

Remark on the Calabi flow with bounded curvature
[abstract] [pdf]
Univ. Iagel. Acta Math. 50 (2013), 107115
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.

(with T. Collins)
The twisted KählerRicci flow
[abstract] [pdf]
to appear in J. Reine Angew. Math.
In this paper we study a generalization of the KählerRicci flow, in which the Ricci form is twisted by a closed, nonnegative (1,1)form. We show that when a twisted KählerEinstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the KählerRicci flow, and it builds on work of TianZhu.

(with T. Collins)
KSemistability for irregular Sasakian manifolds
[abstract] [pdf]
to appear in J. Differential Geom.
We introduce a notion of Ksemistability for Sasakian manifolds. This extends to the irregular case the orbifold Ksemistability of RossThomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily Ksemistable. As an application, we show how one can recover the volume minimization results of MartelliSparksYau, and the Lichnerowicz obstruction of GauntlettMartelliSparksYau from this point of view.

Filtrations and testconfigurations
[abstract] [pdf]
to appear in Math. Ann.
We introduce a strengthening of Kstability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of testconfigurations, 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 bstability.

(with J. Song and B. Weinkove)
The KählerRicci flow on projective bundles
[abstract] [pdf]
Int. Math. Res. Not. 2013, 243257
We study the behaviour of the KählerRicci 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 GromovHausdorff sense to a
metric on the base.

(with D. McFeron)
On the positive mass theorem for manifolds with corners
[abstract]
[pdf]
Comm. Math. Phys. 313 (2012), 425443
We study the positive mass theorem for certain
nonsmooth 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]
Math. Res. Lett. 18 (2011) n. 6., 12711279
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]
Duke Math. J., 161 (2012) n. 8, 14111453
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 ArezzoPacardSinger. We also study the
Kpolystability of these blowups, sharpening a result of
Stoppa in this case. As an application we show that the
blowup of a KahlerEinstein manifold at a point admits a
constant scalar curvature Kahler metric in classes that
make the exceptional divisor small, if it is
Kpolystable with respect to these classes.

(with J. Stoppa)
Relative Kstability of extremal metrics
[abstract]
[pdf]
J. Eur. Math. Soc. 13 (2011) n. 4, 899909
We show that if a polarised manifold admits an extremal
metric then it is Kpolystable relative to a maximal
torus of automorphisms.

(with V. Tosatti)
Regularity of weak solutions of a complex MongeAmpère equation
[abstract]
[pdf]
Analysis & PDE 4 (2011), n. 3, 369378
We prove the smoothness of weak solutions to an elliptic complex MongeAmpère
equation, using the smoothing property of the corresponding parabolic flow.

(with O. Munteanu)
On convergence of the KählerRicci flow
[abstract]
[pdf]
Comm. Anal. Geom. 19 (2011), n. 5, 887904
We study the convergence of the KählerRicci 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), 319331
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 KählerEinstein 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 KählerRicci flow and Kpolystability
[abstract]
[pdf]
Amer. J. Math. 132 (2010), 10771090
We consider the KählerRicci 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 Kpolystable, then the manifold admits a
KählerEinstein 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
Kpolystable.

The Calabi functional on a ruled surface
[abstract]
[pdf]
Ann. Sci. Éc. Norm. Supér. 42 (2009), 837856
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 testconfigurations for toric varieties
[abstract]
[pdf]
J. Differential Geom. 80 (2008), 501523
On a Kunstable 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 HarderNarasimhan
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 testconfigurations, as
predicted by a conjecture of Donaldson.

Extremal metrics and Kstability
[abstract]
[pdf]
Bull. London Math. Soc. 39 (2007), 7684
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
KählerEinstein 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), 15811586
Let G be an Abelian group and let C^G denote
the linear space of all complexvalued 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.
Book
An
Introduction to Extremal Kähler Metrics,
Graduate Studies in Mathematics, AMS
Thesis
The title of my PhD thesis is
Extremal metrics and
Kstability,
supervised by Simon Donaldson.
[abstract]
[pdf]
In this thesis we study the relationship between the existence of
canonical metrics on a complex manifold and stability in the sense of
geometric invariant theory. We introduce a modification of Kstability
of a polarised variety
which we conjecture to be equivalent to the existence of an extremal
metric in the polarisation class.
A variant for a complete extremal metric on the complement of a
smooth divisor is also given. On toric surfaces we prove a
JordanHölder type theorem for decomposing
semistable surfaces into stable pieces.
On a ruled surface we compute the infimum of the
Calabi functional for the unstable polarisations, exhibiting a
decomposition analogous to the HarderNarasimhan filtration of an
unstable vector bundle.
Links