After an introduction to toric symplectic manifolds, I will present an action-angle 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].

[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. 1-24. math.DG/0004122

[2] M. Abreu, Toric Kaehler metrics: cohomogeneity one examples of constant scalar curvature in action-angle coordinates, Journal of Geometry and Symmetry in Physics, 17 (2010), 1-33. 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), 411-454. arXiv:0806.0606

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 Donaldson-Tian-Yau 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 Donaldson-Tian-Yau that this should be equivalent to "K-stability" 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.

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 4-manifold 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 Gieseker-Mumford stability of polarized manifolds. J. Differential Geom., 49(3):577–599, 1998.

T. Mabuchi. K-energy 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. Kähler-Einstein 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 Monge-Ampère equation. I. Comm. Pure Appl. Math., 31(3):339–411, 1978.

The goal of these lectures is to provide a self-contained introduction to the asymptotics of the Bergman kernel and Berezin-Toeplitz quantization. Our approach is based on kernel calculus and the off-diagonal 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: Berezin-Toeplitz quantization
Third lecture: Computation of the Coefficients

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: Berezin-Toeplitz quantization on Kaehler manifolds, J. reine angew. Math., 662 (2012), 1-56. arXiv:1009.5405

X. Ma, G. Marinescu: Berezin-Toeplitz Quantization and its kernel expansion, arXiv:1203.4201

In these lectures I will describe how to associate to Bohr-Sommerfeld, 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, B-S 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 Herman-Kluk propagator.

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 Bohr-Sommerfeld conditions for the Toeplitz operators. Comm. Partial Differential Equations 28 (2003), no. 9-10, 1527–1566.

D. Robert. On the Herman-Kluk semiclassical approximation. Rev. Math. Phys. 22 (2010), no. 10, 1123–1145.