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]

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

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.

• 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