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.
General
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