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.
The Kodaira–Spencer map was originally constructed in the setting of complex manifolds.
gluing the charts together, the idea of deformation theory is to replace these transition maps
by parametrized transition maps
Then, these functions must also satisfy a cocycle condition, which gives a 1-cocycle on
Since the base can be assumed to be a polydisk, this process gives a map between the tangent space of the base to
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
, there is a natural bijection between isomorphisms classes of
the construction of the Kodaira–Spencer map[4] can be done using an infinitesimal interpretation of the cocycle condition.
Recall that a deformation is given by a commutative diagram
is the ring of dual numbers and the vertical maps are flat, the deformation has the cohomological interpretation as cocycles
satisfy the cocycle condition, then they glue to the deformation
One of the original constructions of this map used vector fields in the settings of differential geometry and complex analysis.
[1] 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
{\displaystyle {\begin{aligned}{\frac {\partial f_{ik}^{\alpha }(z_{k},t)}{\partial t}}&={\frac {\partial f_{ij}^{\alpha }(z_{j},t)}{\partial t}}+\sum _{\beta =0}^{n}{\frac {\partial f_{ij}^{\alpha }(z_{j},t)}{\partial f_{jk}^{\beta }(z_{k},t)}}\cdot {\frac {\partial f_{jk}^{\beta }(z_{k},t)}{\partial t}}\\\end{aligned}}}
With a change of coordinates of the part of the previous holomorphic vector field having these partial derivatives as the coefficients, we can write
have a Kodaira-Spencer class constructed cohomologically.
Associated to this deformation is the short exact sequence
gives the short exact sequence
Using derived categories, this defines an element in
Notice this could be generalized to any smooth map
using the cotangent sequence, giving an element in
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
If the two maps in the composition are smooth maps of schemes, then this class coincides with the class in
The Kodaira–Spencer map when considering analytic germs is easily computable using the tangent cohomology in deformation theory and its versal deformations.
gives the long exact sequence
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.
(which is a truncated version of the fundamental triangle) the connecting map of the long exact sequence is the dual of