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[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

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.