Several complex variables and analytic spaces Definition

Kuranishi family

Builds on Deformation of complex structure Kodaira–Spencer map Complex analytic space Zariski tangent space Complex manifold

Definition

Let $X$ be a compact complex manifold. A deformation of $X$ over a pointed complex analytic space $(S, s_0)$ consists of two pieces of data. The first is a proper flat holomorphic map $\varpi \colon \mathcal{X} \to S$ whose fibres are compact complex manifolds. The second is a biholomorphism $X \cong \varpi^{-1}(s_0)$ . This extends the notion of deformation over manifolds. A Kuranishi family (semi-universal deformation) of $X$ is a deformation $\varpi \colon \mathcal{X} \to (S, s_0)$ of $X$ such that:

It is complete: let $\mathcal{Y} \to (T, t_0)$ be any deformation of $X$ . Then there are a neighbourhood $T'$ of $t_0$ and a holomorphic map $f \colon (T', t_0) \to (S, s_0)$ such that $\mathcal{Y}|_{T'} \cong f^* \mathcal{X}$ . The isomorphism is required to be compatible with the identifications of the central fibres with $X$ .
The differential $df_{t_0} \colon T_{t_0} T \to T_{s_0} S$ of any such $f$ is uniquely determined by the deformation $\mathcal{Y}$ .

Remarks

Here $T_{s_0} S$ is the Zariski tangent space of $S$ at $s_0$ . Kuranishi's theorem states that every compact complex manifold has a Kuranishi family, unique up to a non-unique isomorphism of germs. Its Kodaira–Spencer map identifies $T_{s_0} S$ with $H^1(X, \Theta_X)$ , and the germ $(S, s_0)$ is the zero set of a holomorphic map from a neighbourhood of $0$ in $H^1(X, \Theta_X)$ to $H^2(X, \Theta_X)$ . If $H^0(X, \Theta_X) = 0$ , the map $f$ in clause 1 is itself unique, and the family is universal.

Example: $X = \mathbb{P}^n$ has $H^1(X, \Theta_X) = 0$ , so its Kuranishi family lives over a single point: $\mathbb{P}^n$ is rigid.

Non-example: for an elliptic curve $E$ , the trivial family $E \times \Delta \to \Delta$ over a disc is a deformation of $E$ but not complete. Every pullback of it is again trivial, while the family $\C/(\Z + \tau\Z)$ near $\tau_0$ contains fibres not biholomorphic to $E$ , so it is not such a pullback.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Complex analytic space concept
Complex manifold concept
Deformation of complex structure concept
Kodaira–Spencer map concept
Zariski tangent space concept
Kuranishi family definition

Tag	`AX46SNQCAA`
URI	`m2://AX46SNQCAA`
Kind	definition
Writing	W0 stub
MSC	32G05, 32G13
Defines	Kuranishi family
Uses	Deformation of complex structure, Kodaira–Spencer map, Complex analytic space, Zariski tangent space, Complex manifold

Kuranishi family

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here