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[edit]

The Kodaira–Spencer map is[1]

\delta: T_0 S \to H^1(X, TX)

where

  • \mathcal{X} \to S is a smooth proper map between complex spaces[2] (i.e., a deformation of the special fiber X = \mathcal{X}_0.)
  • \delta is the connecting homomorphism obtained by taking a long exact cohomology sequence of the surjection T \mathcal{X}|_X \to T_0 S \otimes \mathcal{O}_X whose kernel is the tangent bundle TX.

If v is in T_0 S, then its image \delta(v) is called the Kodaira–Spencer class of v.

The basic fact is: there is a natural bijection between isomorphisms classes of \mathcal{X} \to S = \operatorname{Spec}(\mathbb{C}[t]/t^2) and H^1(X, TX).

References[edit]

  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.