Kodaira

In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations

In the theory of nonsingular projective surfaces, surfaces are first classified into four classes according to their values of Kodaira dimensions. To be more precise, let V be a non- singular projective surface defined over an algebraically closed field k of characteristic zero.

The first instances of deformation theory were given by Kodaira and Spencer for complex structures and by Gerstenhaber for associative algebras. Since then, deformation theory has been applied as a useful tool in the study of many other mathematical structures, and even today it plays an important role in many developments of modern mathematics.