Metrics of Special Holonomy and Gluing Theorems

This book describes applications of the PDE methods to the construction and study of Ricci-flat metrics with special holonomy. Particular attention is paid to Ricci-flat Kahler (Calabi-Yau) structures on complex manifolds and hyper-Kahler structures on K3 surfaces. Complex manifolds are also an object of study in algebraic geometry and special consideration is given to the interplay between some well-known algebraic varieties (K3 surfaces, Fano threefolds, for example) and differential-geometric structures of special holonomy. The interplay between the gluing techniques, Calabi-Yau theory and algebraic geometry is further illuminated by the connected sum construction of compact 7-dimensional manifolds with holonomy G2.