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Kodaira–Spencer map

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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

δ : T 0 S → H 1 ( X , T X )

where

  • X → S is a smooth proper map between complex spaces (i.e., a deformation of the special fiber X = X 0 .)
  • δ is the connecting homomorphism obtained by taking a long exact cohomology sequence of the surjection T X | X → T 0 S ⊗ O X whose kernel is the tangent bundle T X .

    • If v is in T 0 S , then its image δ ( v ) is called the Kodaira–Spencer class of v.

    The basic fact is: there is a natural bijection between isomorphisms classes of X → S = Spec ⁡ ( C [ t ] / t 2 ) and H 1 ( X , T X ) .

    References

    Kodaira–Spencer map Wikipedia
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