Deformation of complex structure

Summary

A deformation of the complex structure of a compact complex manifold $X$ is a holomorphic family of compact complex manifolds. The family is parametrized by a base, and its fibre over a chosen base point is $X$. By Ehresmann's theorem all fibres are diffeomorphic, so the family varies only the complex structure on a fixed smooth manifold. Kodaira and Spencer showed that first-order deformations are measured by $H^1(X, \Theta_X)$, where $\Theta_X$ is the sheaf of holomorphic vector fields, and obstructions lie in $H^2(X, \Theta_X)$. The theory is the analytic part of deformation theory and leads to local moduli spaces.