The K¨ahler geometry of toric manifolds
                            Vestislav Apostolov
                Email address: apostolov.vestislav@uqam.ca
                                                       Contents
                        Preface                                                                  5
                          Acknowledgements                                                       5
                        Chapter 1.  Delzant Theory                                               7
                          1.  Hamiltonian group actions on symplectic manifolds                  7
                          2.  Hamiltonian actions of tori                                       10
                          3.  The complex projective space                                      12
                          4.  Toric symplectic manifolds from Delzant polytopes                 17
                          5.  Toric complex varieties from Delzant polytopes. Fans              19
                          6.  Equivariant blow-up                                               20
                          7.  Polarized projective toric varieties                              22
                          8.  Toric orbifolds                                                   24
                        Chapter 2.  Abreu–Guillemin Theory                                      27
                          1.  The orbit space of a toric manifold                               27
                          2.  Toric K¨ahler metrics: local theory                               27
                          3.  The scalar curvature                                              30
                          4.  Symplectic versus complex: the Legendre transform                 31
                          5.  The canonical toric K¨ahler metric                                33
                          6.  Toric K¨ahler metrics: compactification                            34
                        Chapter 3.  The Calabi Problem and Donaldson’s theory                   37
                          1.  The Calabi problem on a toric manifold                            37
                          2.  Donaldson–Futaki invariant                                        40
                          3.  Uniqueness                                                        40
                          4.  K-stability                                                       41
                          5.  Toric test configurations                                          42
                          6.  Uniform K-stability                                               45
                          7.  Existence: an overview                                            48
                        Bibliography                                                            59
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