Calabi-Yau Manifolds and Related Geometries

• I. Riemannian Holonomy Groups and Calibrated Geometry.- 1 Introduction.- 2 Introduction to Holonomy Groups.- 3 Berger’s Classification of Holonomy Groups.- 4 Kahler Geometry and Holonomy.- 5 The Calabi Conjecture.- 6 The Exceptional Holonomy Groups.- 7 Introduction to Calibrated Geometry.- 8 Calibrated Submanifolds in ?n.- 9 Constructions of SL m-folds in ?m.- 10 Compact Calibrated Submanifolds.- 11 Singularities of Special Lagrangian m-folds.- 12 The SYZ Conjecture, and SL Fibrations.- II. Calabi-Yau Manifolds and Mirror Symmetry.- 13 Introduction.- 14 The Classical Geometry of Calabi-Yau Manifolds.- 15 Kahler Moduli and Gromov-Witten Invariants.- 16 Variation and Degeneration of Hodge Structures.- 17 A Mirror Conjecture.- 18 Mirror Symmetry in Practice.- 19 The Strominger-Yau-Zaslow Approach to Mirror Symmetry.- III. Compact Hyperkähler Manifolds.- 20 Introduction.- 21 Holomorphic Symplectic Manifolds.- 22 Deformations of Complex Structures.- 23 The Beauville-Bogomolov Form.- 24 Cohomology of Compact Hyperkähler Manifolds.- 25 Twistor Space and Moduli Space.- 26 Projectivity of Hyperkähler Manifolds.- 27 Birational Hyperkähler Manifolds.- 18 The (Birational) Kahler Cone.- References.

From the reviews: "Summer schools in Nordfjordeid, Norway, have been organized regularly since 1996. Topics vary each year but there are always three series of lectures by invited experts with evening exercises. [...] The themes of all three contributions are interrelated and together they give a nice introduction into a very interesting field of research on the border between mathematics and physics. l would like to strongly recommend the book to anybody interested in the topic." (jbu) European Mathematical Society Newsletter, Sept. 2004, p. 44 "[...] This book is an excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kahler manifolds, exceptional holonomy and mirror symmetry. [...] This is an excellent and useful book. The different chapters mostly fit together reasonably well [...]. It will be popular with anyone trying to learn about Kahler manifolds with c1=0 and mirror symmetry." Richard P. Thomas, Mathematical Reviews, Clippings from Issue 2004c "Summer schools in Nordfjordeid, Norway, have been organized regularly since 1996. ... The school held in June 2001 was devoted to recent interaction between differential and algebraic geometry. The book consists of notes written by lecturers of the corresponding three series of lectures. ... they give a nice introduction into a very interesting field of research on the border between mathematics and physics. I would like to strongly recommend the book to anybody interested in the topic." (EMS, September, 2004) "This book is an excellent introduction to current research in the geometry of Calabi-Yau manifolds, hyper-Kahler manifolds, exceptional holonomy and mirror symmetry. ... the choice of topics is a sensible one. ... This is an excellent and useful book. ... It will be popular with anyone trying to learn about Kahler manifolds with c1 = 0 and mirror symmetry." (Richard P. Thomas, Mathematical Reviews, 2004 c)