Kodaira–Spencer map
In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X. == Definition == === Historical motivation === The Kodaira–Spencer map was originally constructed in the setting of complex manifolds. Given a complex analytic manifold M {\displaystyle M} with charts U i {\displaystyle U_{i}} and biholomorphic maps f j k {\displaystyle f_{jk}} sending z k → z j = ( z j 1 , … , z j n ) {\displaystyle z_{k}\to z_{j}=(z_{j}^{1},\ldots ,z_{j}^{n})} gluing the charts together, the idea of deformation theory is to replace these transition maps f j k ( z k ) {\displaystyle f_{jk}(z_{k})} by parametrized transition maps f j k ( z k , t 1 , … , t k ) {\displaystyle f_{jk}(z_{k},t_{1},\ldots ,t_{k})} over some base B {\displaystyle B} (which could be a real manifold) with coordinates t 1 , … , t k {\displaystyle t_{1},\ldots ,t_{k}} , such that f j k ( z k , 0 , … , 0 ) = f j k ( z k ) {\displaystyle f_{jk}(z_{k},0,\ldots ,0)=f_{jk}(z_{k})} .