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Kodaira–Spencer map

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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

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Historical motivation

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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.[1]

Original definition

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More formally, the Kodaira–Spencer map is[2]

where

  • is a smooth proper map between complex spaces[3] (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

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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

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Using infinitesimals

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Cocycle condition for deformations

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Over characteristic the construction of the Kodaira–Spencer map[4] 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

Conversion to cocycles of vector fields

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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

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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 . 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

With a change of coordinates of the part of the previous holomorphic vector field having these partial derivatives as the coefficients, we can write

Hence we can write up the equation above as the following equation of vector fields

Rewriting this as the vector fields

where

gives the cocycle condition. Hence has an associated class in from the original deformation of .

In scheme theory

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Deformations of a smooth variety[5]

have 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 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

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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[6] (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

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With analytic germs

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The Kodaira–Spencer map when considering analytic germs is easily computable using the tangent cohomology in deformation theory and its versal deformations.[7] For example, given the germ of a polynomial , its space of deformations can be given by the module

For example, if then its versal 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

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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

(which is a truncated version of the fundamental triangle) 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 .[8] Then, the Kodaira–Spencer map sends a deformation

to the element .

See also

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References

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  1. ^ a b Kodaira (2005). Complex Manifolds and Deformation of Complex Structures. Classics in Mathematics. pp. 182–184, 188–189. doi:10.1007/b138372. ISBN 978-3-540-22614-7.
  2. ^ Huybrechts 2005, 6.2.6.
  3. ^ The main difference between a complex manifold and a complex space is that the latter is allowed to have a nilpotent.
  4. ^ Arbarello; Cornalba; Griffiths (2011). Geometry of Algebraic Curves II. Grundlehren der mathematischen Wissenschaften, Arbarello,E. Et al: Algebraic Curves I, II. Springer. pp. 172–174. ISBN 9783540426882.
  5. ^ Sernesi. "An overview of classical deformation theory" (PDF). Archived (PDF) from the original on 2020-04-27.
  6. ^ Illusie, L. Complexe cotangent ; application a la theorie des deformations (PDF). Archived from the original (PDF) on 2020-11-25. Retrieved 2020-04-27.
  7. ^ Palamodov (1990). "Deformations of Complex Spaces". Several Complex Variables IV. Encyclopaedia of Mathematical Sciences. Vol. 10. pp. 138, 130. doi:10.1007/978-3-642-61263-3_3. ISBN 978-3-642-64766-6.
  8. ^ Talpo, Mattia; Vistoli, Angelo (2011-01-30). "Deformation theory from the point of view of fibered categories". pp. 25, exercise 3.25. arXiv:1006.0497 [math.AG].