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  (which could be a real manifold) 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
 is a smooth proper map between complex spaces (i.e., a deformation of the special fiber .) is the connecting homomorphism obtained by taking a long exact cohomology sequence of the surjection  whose kernel is the tangent bundle . 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 diagramwhere  is the ring of dual numbers and the vertical maps are flat, the deformation has the cohomological interpretation as cocycles  on  whereIf the  satisfy the cocycle condition, then they glue to the deformation . This can be read asUsing the properties of the dual numbers, namely , we haveandhence the cocycle condition on  is the following two rules

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 fieldwhich 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 asThen, the derivative of  with respect to  can be calculated from the previous equation asNote because  and , then the derivative reads as With a change of coordinates of the part of the previous holomorphic vector field having these partial derivatives as the coefficients, we can writeHence we can write up the equation above as the following equation of vector fieldsRewriting this as the vector fieldswheregives the cocycle condition. Hence  has an associated class in  from the original deformation  of . In scheme theory 
Deformations of a smooth varietyhave a Kodaira-Spencer class constructed cohomologically. Associated to this deformation is the short exact sequence(where ) which when tensored by the -module  gives the short exact sequenceUsing derived categories, this defines an element ingeneralizing 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 topoiThen, associated to this composition is a distinguished triangleand this boundary map forms the Kodaira–Spencer map (or cohomology class, denoted ). 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 moduleFor example, if  then its versal deformations is given byhence an arbitrary deformation is given by . Then for a vector , which has the basisthere 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 triangleThen, applying  gives the long exact sequenceRecall that there is the isomorphismfrom 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 isomorphismsThe last isomorphism comes from the isomorphism , and a morphism in  send giving the desired isomorphism. From the cotangent sequence(which is a truncated version of the fundamental triangle) the connecting map of the long exact sequence is the dual of , giving the isomorphismNote this computation can be done by using the cotangent sequence and computing . Then, the Kodaira–Spencer map sends a deformationto the element . See also 
Deformation theory
Cotangent complex
Schlessinger's theorem
characteristic linear system of an algebraic family of curves
Gauss–Manin connection
Derived category
Ext functor

References 

Mathoverflow post relating deformations to the jacobian ring. Algebraic geometry