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

Historical motivation

The Kodaira–Spencer map was originally constructed in the setting of complex manifolds. Given a complex analytic manifold with charts and biholomorphic maps sending gluing the charts together, the idea of deformation theory is to replace these transition maps by parametrized transition maps over some base with coordinates, such that. This means the parameters deform the complex structure of the original complex manifold. Then, these functions must also satisfy a cocycle condition, which gives a 1-cocycle on with values in its tangent bundle. Since the base can be assumed to be a polydisk, this process gives a map between the tangent space of the base to called the Kodaira–Spencer map.

Original definition

More formally, the Kodaira–Spencer map is
where
If is in, then its image is called the Kodaira–Spencer class of.

Remarks

Because deformation theory has been extended to multiple other contexts, such as deformations in scheme theory, or ringed topoi, there are constructions of the Kodaira–Spencer map for these contexts.
In the scheme theory over a base field of characteristic, there is a natural bijection between isomorphisms classes of and.

Constructions

Using infinitesimals

Cocycle condition for deformations

Over characteristic the construction of the Kodaira–Spencer map can be done using an infinitesimal interpretation of the cocycle condition. If we have a complex manifold covered by finitely many charts with coordinates and transition functions
where
Recall that a deformation is given by a commutative diagram
where is the ring of dual numbers and the vertical maps are flat, the deformation has the cohomological interpretation as cocycles on where
If the satisfy the cocycle condition, then they glue to the deformation. This can be read as
Using the properties of the dual numbers, namely, we have
and
hence the cocycle condition on is the following two rules

  2. Conversion to cocycles of vector fields

The cocycle of the deformation can easily be converted to a cocycle of vector fields as follows: given the cocycle we can form the vector field
which is a 1-cochain. Then the rule for the transition maps of gives this 1-cochain as a 1-cocycle, hence a class.

Using vector fields

One of the original constructions of this map used vector fields in the settings of differential geometry and complex analysis. Given the notation above, the transition from a deformation to the cocycle condition is transparent over a small base of dimension one, so there is only one parameter. Then, the cocycle condition can be read as
Then, the derivative of with respect to can be calculated from the previous equation as
Note because and, then the derivative reads as
If we use write a holomorphic vector field, having these partial derivatives as the coefficients, then because
we get the following equation of vector fields
Rewriting this as the vector fields
where
gives the cocycle condition. Hence this has associated a class in from a deformation.

In scheme theory

Deformations of a smooth variety
have a Kodaira-Spencer class constructed cohomologically. Associated to this deformation is the short exact sequence
which when tensored by the -module gives the short exact sequence
Using derived categories, this defines an element in
generalizing the Kodaira–Spencer map. Notice this could be generalized to any smooth map in using the cotangent sequence, giving an element in.

Of ringed topoi

One of the most abstract constructions of the Kodaira–Spencer maps comes from the cotangent complexes associated to a composition of maps of ringed topoi
Then, associated to this composition is a distinguished triangle
and this boundary map forms the Kodaira–Spencer map. If the two maps in the composition are smooth maps of schemes, then this class coincides with the class in.

Examples

With analytic germs

The Kodaira–Spencer map when considering analytic germs is easily computable using the tangent cohomology in deformation theory and its versal deformations. For example, given the germ of a polynomial, its space of deformations can be given by the module
For example, if then its veral deformations is given by
hence an arbitrary deformation is given by. Then for a vector, which has the basis
there the map sending

On affine hypersurfaces with the cotangent complex

For an affine hypersurface over a field defined by a polynomial, there is the associated fundamental triangle
Then, applying gives the long exact sequence
Recall that there is the isomorphism
from general theory of derived categories, and the ext group classifies the first-order deformations. Then, through a series of reductions, this group can be computed. First, since is a free module,. Also, because, there are isomorphisms
The last isomorphism comes from the isomorphism, and a morphism in
send
giving the desired isomorphism. From the cotangent sequence
the connecting map of the long exact sequence is the dual of, giving the isomorphism
Note this computation can be done by using the cotangent sequence and computing. Then, the Kodaira–Spencer map sends a deformation
to the element.