Canonical metrics in Kähler geometry. Notes taken by Meike Akveld.

*(English)*Zbl 0978.53002
Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser. 100 p. (2000).

Complex differential geometry has developed over the last two decades very intensively. Among possible reasons one can find many applications, including the use of Calabi-Yau spaces in superstring theory. This monograph includes an essentially self-contained introduction to the theory of canonical Kähler metrics on complex manifolds. It includes also some advanced topics in complex differential geometry which are hard to be found elsewhere.

The monograph consists of seven chapters. The first chapter deals with fundamental notions in Kähler geometry: Kähler metrics and curvature of Kähler metrics, including the most important examples of these spaces. To characterize Kähler manifolds of constant curvature in terms of Chern numbers in Chapter 2 the author introduces the Calabi functional on the space of Kähler metrics and gives a brief review of Chern classes. Chapter 3 deals with a holomorphic invariant for Kähler manifolds, so called Calabi-Futaki invariants. Here a slightly different approach from the original one is taken. The author computes these invariants for three explicit Kähler manifolds and states a localization formula for these ones. The goal of Chapter 4 is to prove that the scalar curvature of Kähler metrics is a moment map with respect to \(Symp(M)\), the group of symplectomorphisms of a compact Kähler manifold. The proof of the Calabi-Yau theorem on the existence of Kähler Ricci-flat metrics as well as a unique Kähler-Einstein metric on a compact Kähler manifold with \(c_1(M)<0\) are given in Chapter 5. Recent progress on Kähler-Einstein metrics with positive scalar curvature is presented in Chapter 6. In Section 7 the author discusses some applications of the theorems from previous chapters and gives also some generalizations of previous results. The book includes also some open questions and conjectures as well as examples to clarify the corresponding subject.

The monograph consists of seven chapters. The first chapter deals with fundamental notions in Kähler geometry: Kähler metrics and curvature of Kähler metrics, including the most important examples of these spaces. To characterize Kähler manifolds of constant curvature in terms of Chern numbers in Chapter 2 the author introduces the Calabi functional on the space of Kähler metrics and gives a brief review of Chern classes. Chapter 3 deals with a holomorphic invariant for Kähler manifolds, so called Calabi-Futaki invariants. Here a slightly different approach from the original one is taken. The author computes these invariants for three explicit Kähler manifolds and states a localization formula for these ones. The goal of Chapter 4 is to prove that the scalar curvature of Kähler metrics is a moment map with respect to \(Symp(M)\), the group of symplectomorphisms of a compact Kähler manifold. The proof of the Calabi-Yau theorem on the existence of Kähler Ricci-flat metrics as well as a unique Kähler-Einstein metric on a compact Kähler manifold with \(c_1(M)<0\) are given in Chapter 5. Recent progress on Kähler-Einstein metrics with positive scalar curvature is presented in Chapter 6. In Section 7 the author discusses some applications of the theorems from previous chapters and gives also some generalizations of previous results. The book includes also some open questions and conjectures as well as examples to clarify the corresponding subject.

Reviewer: Neda Bokan (Beograd)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

58E11 | Critical metrics |

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53D20 | Momentum maps; symplectic reduction |