Gábor Székelyhidi
Columbia
University
Department
of Mathematics, Rm 629
2990 Broadway
New York, NY
10027
email : gabor at math.columbia.edu
I am a Ritt Assistant Professor at Columbia University, interested in
algebraic/differential geometry. My current research problem is
understanding when extremal metrics exist on a polarized algebraic
variety, and related questions.
Teaching
Calculus III - Spring 2010 - Section 2
Calculus III - Spring 2010 - Section 3
Papers
-
(with J. Stoppa)
Relative K-stability of extremal metrics
[abstract]
[pdf]
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]
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]
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]
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]
to appear in Amer. J. Math.
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.
Thesis
The title of my PhD thesis is
Extremal metrics and
K-stability,
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 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.
Links
