University of Notre Dame
South Bend, IN 46556
email : gszekely at nd.edu
I am an Associate Professor of
Mathematics at the University of Notre Dame,
algebraic/differential geometry. My current research problem is
understanding when extremal metrics exist on a polarized algebraic
variety, and related questions.
Basic Complex Analysis I
office hours on Wednesdays 9--10:30am, or by appointment.
Problem sets: 1
(with T. Collins)
Sasaki-Einstein metrics and K-stability
We show that a polarized affine variety admits a Ricci flat Kähler cone metric, if it is K-stable. This generalizes Chen-Donaldson-Sun's solution of the Yau-Tian-Donaldson conjecture to Kähler cones, or equivalently, Sasakian manifolds. As an application we show that the five-sphere admits infinitely many families of Sasaki-Einstein metrics.
(with B. Weinkove)
On a constant rank theorem for nonlinear elliptic PDEs
to appear in Dicrete Contin. Dyn. Syst. Ser. B
We give a new proof of Bian-Guan'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
(with V. Datar)
Kähler-Einstein metrics along the smooth continuity
to appear in GAFA
We show that if a Fano manifold M is K-stable with respect to special
degenerations equivariant under a compact group of automorphisms, then M
admits a Kähler-Einstein metric. This is a strengthening of the solution of
the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-Donaldson-Sun, and
can be used to obtain new examples of Kähler-Einstein manifolds. We also give
analogous results for twisted Kähler-Einstein metrics and Kähler-Ricci
(with V. Tosatti, B. Weinkove)
Gauduchon metrics with prescribed volume form
We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.
Fully non-linear elliptic equations on compact Hermitian manifolds
to appear in J. Differential Geom.
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, 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 J-flow on toric manifolds
to appear in J. Differential Geom.
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. On toric manifolds we give a numerical characterization of when the J-flow 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
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 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 has
been established. We then turn to the question of what one expects
when no extremal metric exists.
The partial C0-estimate along the continuity
to appear in J. Amer. Math. Soc.
We prove that the partial C0-estimate holds for metrics along Aubin's continuity method for finding Kähler-Einstein metrics, confirming a special case of a conjecture due to Tian. We use the method developed in recent work of Chen-Donaldson-Sun on the analogous problem for conical Kähler-Einstein metrics.
(with M. Lejmi)
The J-flow and stability
to appear in Advances in Math.
We study the J-flow from the point of view of an algebro-geometric 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 Fang-Lai is reflected by the optimal destabilizer. Finally we prove a general existence result on complex tori.
Blowing up extremal Kähler manifolds II
to appear in Invent. Math.
This is a continuation of the work of Arezzo-Pacard-Singer and the author on
blowups of extremal Kähler manifolds. We prove the conjecture stated in ,
and we relate this result to the K-stability 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 K-stable, as long as the exceptional divisor is sufficiently
A remark on conical Kähler-Einstein metrics
Math. Res. Lett. 20 (2013) n. 3., 581--590
We give some non-existence results for Kähler-Einstein 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 K-stability.
Remark on the Calabi flow with bounded curvature
Univ. Iagel. Acta Math. 50 (2013), 107--115
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 Kähler-Ricci flow
to appear in J. Reine Angew. Math.
In this paper we study a generalization of the Kähler-Ricci flow, in which the Ricci form is twisted by a closed, non-negative (1,1)-form. We show that when a twisted Kähler-Einstein metric exists, then this twisted flow converges exponentially. This generalizes a result of Perelman on the convergence of the Kähler-Ricci flow, and it builds on work of Tian-Zhu.
(with T. Collins)
K-Semistability for irregular Sasakian manifolds
to appear in J. Differential Geom.
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
to appear in Math. Ann.
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
Int. Math. Res. Not. 2013, 243--257
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
Comm. Math. Phys. 313 (2012), 425--443
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
(with Renjie Feng)
Periodic solutions of Abreu's equation
Math. Res. Lett. 18 (2011) n. 6., 1271--1279
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
On blowing up extremal Kähler manifolds
Duke Math. J., 161 (2012) n. 8, 1411--1453
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
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
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
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
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
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
The Calabi functional on a ruled surface
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
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
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
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.
Introduction to Extremal Kähler Metrics
Graduate Studies in Mathematics, AMS
The title of my PhD thesis is Extremal metrics and
supervised by Simon Donaldson.
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 K-stability
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
Jordan-Hölder 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 Harder-Narasimhan filtration of an
unstable vector bundle.