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.
The Kodaira–Spencer map is
- is a smooth proper map between complex spaces (i.e., a deformation of the special fiber .)
- is the connecting homomorphism obtained by taking a long exact cohomology sequence of the surjection whose kernel is the tangent bundle .
If is in , then its image is called the Kodaira–Spencer class of v.
The basic fact is: there is a natural bijection between isomorphisms classes of and .
- Huybrechts 2005, 6.2.6.
- The main difference between a complex manifold and a complex space is that the latter is allowed to have a nilpotent.
- Huybrechts, Daniel (2005). Complex Geometry: An Introduction. Springer. ISBN 3-540-21290-6.
- Kodaira, Kunihiko (1986), Complex manifolds and deformation of complex structures, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 283, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96188-0, MR 815922
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