Kodaira–Spencer map

Summary

The Kodaira–Spencer map of a deformation measures how fast the complex structure of the fibre changes. It sends a tangent vector of the base at $t$ to a class in $H^1(X_t, \Theta_{X_t})$. Here $\Theta_{X_t}$ is the sheaf of holomorphic vector fields on the fibre $X_t$. It vanishes identically for a trivial family, and it is the basic tool to decide whether a family is complete or effectively parametrized. It is the first-order part of the deformation theory of compact complex manifolds.