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

The Kodaira–Spencer map is[1]

where

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 .

References

  1. Huybrechts 2005, 6.2.6.
  2. The main difference between a complex manifold and a complex space is that the latter is allowed to have a nilpotent.


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