KodairaSpencerAbbildung

Kodaira-Spencer theory is a foundational framework in complex geometry for the deformation of complex structures on manifolds and complex spaces. Named after Kunihiko Kodaira and Donald C. Spencer, the theory was developed in the 1950s to formalize how complex structures vary in families and how infinitesimal deformations are classified.

For a compact complex manifold X and a holomorphic family f: X → S with fiber X_s, the Kodaira-Spencer map at s is a linear map from the tangent space T_s S to H^1(X_s, T_{X_s}). It is induced by the connecting morphism in the exact sequence 0 → T_{X/S} → T_X → f^* T_S → 0 and measures how the complex structure on X_s varies infinitesimally. An infinitesimal deformation of X corresponds to a Kodaira-Spencer class in H^1(X, T_X). In this framework, H^1(X, T_X) is identified with the tangent space to the moduli space of complex structures at [X], under appropriate identifications.

Obstructions to extending first-order deformations to higher order live in H^2(X, T_X). While deformations can be obstructed in general, there are important cases in which they are unobstructed, such as Calabi-Yau manifolds, where the Bogomolov-Tian-Todorov theorem applies and deformations of the complex structure form a smooth moduli space.

The theory also integrates the variation of Hodge structure through the period map and provides tools for constructing and analyzing families of complex manifolds. It has broad applications to moduli problems and to the study of examples such as complex tori, K3 surfaces, and higher-dimensional Calabi-Yau varieties.