KodairaSpencer

Kodaira–Spencer theory, named after Kunihiko Kodaira and Donald C. Spencer, is a foundational framework in complex geometry for studying how complex structures on a differentiable manifold vary in families. Developed in the 1950s and 1960s, it provides tools to analyze deformations of complex manifolds and to describe how a complex structure changes under small perturbations.

In the context of a smooth family f: X -> S of complex manifolds, the Kodaira–Spencer map at a point s in S is a linear map from the tangent space T_s S to the Dolbeault cohomology group H^1(X_s, T_{X_s}). Its image encodes the infinitesimal variation of the complex structure of the fiber X_s, and the corresponding cohomology class is called the Kodaira–Spencer class. Infinitesimal deformations are controlled by H^1(X_s, T_{X_s}), while obstructions to extending them lie in H^2(X_s, T_{X_s}). The theory often uses the sheaf of holomorphic vector fields T_{X} and Dolbeault techniques to translate geometric questions into cohomological ones.

Kodaira–Spencer theory underpins the study of moduli spaces of complex structures, including the existence of versal or universal deformation spaces. It has applications in algebraic geometry and complex manifolds, such as the study of Calabi–Yau manifolds where obstructions can vanish (the Tian–Todorov unobstructedness result). The framework also interacts with related deformation theories, Kodaira–Spencer differential graded Lie algebras, and mirror symmetry.