Kaehler geometry of toric symplectic manifolds in actionangle coordinates, Miguel Abreu.
Outline
After an introduction
to toric symplectic manifolds, I will present an actionangle
coordinates approach to toric Kaehler geometry [1] and discuss two of
its applications: toric Kaehler metrics of constant scalar
curvature [2] and geometric quantization of compact toric symplectic manifolds [3].
Literature
[1] M. Abreu, Kaehler
geometry of toric manifolds in symplectic coordinates, in "Symplectic
and Contact Topology: Interactions and Perspectives" (eds. Y.
Eliashberg, B. Khesin and F. Lalonde), Fields Institute Communications
35, American Mathematical Society, 2003, pp. 124. math.DG/0004122
[2] M. Abreu, Toric Kaehler metrics: cohomogeneity one examples of
constant scalar curvature in actionangle coordinates, Journal of
Geometry and Symmetry in Physics, 17 (2010), 133. arXiv:0912.0491
[3] T. Baier, C. Florentino, J. Mourão, and J. P. Nunes, Toric Kähler
metrics seen from infinity, quantization and compact tropical amoebas,
J. Differential Geom. 89 (2011), 411454. arXiv:0806.0606
Quantisation and canonical metrics in Kähler geometry, Joel Fine.
Lecture notes.
Outline
The
aim of these lectures is to provide an introduction to the study of
canonical metrics in Kähler geometry and the interaction between this
field and that of quantisation of Kähler manifolds. The goal will to be
to provide an overview of a large and rapidly increasing area of
research. I will give no proofs, focusing instead on explaining why one
might expect the various results to be true, leaving it to the audience
to fill in the details from the literature.
Lecture 1. Canonical Kähler metrics.
I will begin with the questions posed by Calabi which instigated the
entire field of canonical Kähler metrics. Calabi was interested in
finding the "best" Kähler metric in a given Kähler class. I will discus
Calabi's various suggestions and then I will attempt to give an
overview of the current state of research: conditions under which
"best" metrics are known to exist and some of the obstructions to their
existence in general. To tackle such a large subject in 90 minutes
would be hopeless, instead I will pick and choose certain results to
focus on, leaving it to the audience to discover the rest for
themselves from the literature suggested below.
Lecture 2. From balanced projective embeddings to the DonaldsonTianYau conjecture.
Here we will seemingly change topic completely and consider instead
projectively equivalent embeddings of a given complex submanifold of
projective space. I will explain how to define a best such embedding,
called "balanced" and the theorem of Luo and Zhang that a balanced
embedding exists if and only if a certain algebrogeometric criterion,
called "Chow stability", is satisfied. From here I will try to draw an
analogy to the problem of finding a constant scalar curvature Kähler
metric in a given integral Kähler class and the conjecture of
DonaldsonTianYau that this should be equivalent to "Kstability" of
the corresponding polarised variety.
Lecture 3. Constant scalar curvature Kähler metrics as the classical limit of balanced embeddings.
In this lecture I will explain the theorem of Donaldson that under
Kähler quantisation, constant scalar curvature Kähler metrics are the
classical limit of balanced embeddings. I will outline the (less
technical parts of the) proof and then go on to discuss other
subsequent results in the same vein. If there is time I will finish by
mentioning some open problems and promising directions of research in
this area.
Literature
I do not expect
people to have read any of this material in advance (although it would
of course help). Instead, a small subset of this is what one could read
in order to fill out the necessarily sketchy story I will tell during
the lectures. I should also stress that this list is extremely far from
being representative; such a list would be pages long.
Surveys
D. H. Phong and J. Sturm. Lectures on Stability and Constant Scalar Curvature. arxiv:0801.4179
R. P. Thomas. Notes on GIT and symplectic reduction for bundles and varieties. arXiv:math/0512411
G. Tian. Canonical metrics in Kähler geometry. Lectures in Mathematics
ETH Zürich. Birkhäuser Verlag, Basel, 2000. Notes taken by Meike Akveld.
Research articles
E. Calabi. Extremal Kähler metrics. In Seminar on Differential
Geometry, volume 102 of Ann. of Math. Stud., pages 259–290. Princeton
Univ. Press, Princeton, N.J., 1982.
E. Calabi. Extremal Kähler metrics. II. In Differential geometry and complex analysis, pages 95–114. Springer, Berlin, 1985.
S. K. Donaldson. Remarks on gauge theory, complex geometry and
4manifold topology. In Fields Medallists’ lectures, volume 5 of World
Sci. Ser. 20th Century Math., pages 384–403. World Sci. Publishing,
River Edge, NJ, 1997.
S. K. Donaldson. Scalar curvature and projective embeddings. I. J. Differential Geom., 59(3):479–522, 2001.
S. K. Donaldson. Scalar curvature and stability of toric varieties. J. Differential Geom., 62(2):289–349, 2002.
H. Luo. Geometric criterion for GiesekerMumford stability of polarized manifolds. J. Differential Geom., 49(3):577–599, 1998.
T. Mabuchi. Kenergy maps integrating Futaki invariants. Tohoku Mathematical Journal, 38:575–593, 1986.
G. Tian. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom., 32(1):99–130, 1990.
G. Tian. KählerEinstein metrics with positive scalar curvature. Invent. Math., 130(1):1–37, 1997.
S.T. Yau. On the Ricci curvature of a compact Kähler manifold and the
complex MongeAmpère equation. I. Comm. Pure Appl. Math.,
31(3):339–411, 1978.
Bergman kernels and BerezinToeplitz quantization, Xiaonan Ma.
Outline
The goal of these
lectures is to provide a selfcontained introduction to the asymptotics
of the Bergman kernel and BerezinToeplitz quantization. Our approach
is based on kernel calculus and the offdiagonal asymptotic expansion
of the Bergman kernel. This method allows not only to derive the
asymptotic expansions of the Toeplitz operators but also to calculate
the first coefficients of the various expansions. Since the formulas
for the coefficients encode geometric data of the manifold and
prequantum bundle they found extensive and deep applications in the
study of Kaehler manifolds.
First lecture: Asymptotic of the Bergman kernel
Second lecture: BerezinToeplitz quantization
Third lecture: Computation of the Coefficients
Literature
X. Ma, G. Marinescu:
Holomorphic Morse inequalities and Bergman kernels, Progress in
Mathematics, vol. 254, Birkhauser Boston Inc., Boston, MA, 2007, 422 pp.
X. Ma, G. Marinescu: Toeplitz operators on symplectic manifolds, J. Geom. Anal. 18 (2008), no. 2, 565–611. arXiv:0806.2370
X. Ma, G. Marinescu: BerezinToeplitz quantization on Kaehler
manifolds, J. reine angew. Math., 662 (2012), 156. arXiv:1009.5405
X. Ma, G. Marinescu: BerezinToeplitz Quantization and its kernel expansion, arXiv:1203.4201
Quantization of Lagrangian Submanifolds, Alejandro Uribe.
Outline
In these lectures I will describe how to associate to BohrSommerfeld, weighted, lagrangian submanifolds of a quantized Kähler manifold sequences of vectors in the holomorphic sections of powers of the line bundle, with very good semiclassical properties. I will also discuss applications of this construction, and mention open problems.
First lecture: The setting, BS lagrangian submanifolds, and their quantization. Examples. Relationship with the ``quantization commutes with reduction" theme.
Second lecture: Symbolic matters and general estimates
Third lecture: Applications: Construction of quasimodes, and generalizing the HermanKluk propagator.
Literature
D. Borthwick, T. Paul, A. Uribe. Legendrian distributions with applications to relative Poincaré series. Invent. Math. 122 (1995), no. 2, 359–402.
L. Charles. Quasimodes and BohrSommerfeld conditions for the Toeplitz operators. Comm. Partial Differential Equations 28 (2003), no. 910, 1527–1566.
D. Robert. On the HermanKluk semiclassical approximation. Rev. Math. Phys. 22 (2010), no. 10, 1123–1145.
