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]
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 .
Remarksedit
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 .
Constructionsedit
Using infinitesimalsedit
Cocycle condition for deformationsedit
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 fieldsedit
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 fieldsedit
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 .
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 topoiedit
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
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 .
Examplesedit
With analytic germsedit
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 complexedit
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
^ abKodaira (2005). Complex Manifolds and Deformation of Complex Structures. Classics in Mathematics. pp. 182–184, 188–189. doi:10.1007/b138372. ISBN978-3-540-22614-7.
^The main difference between a complex manifold and a complex space is that the latter is allowed to have a nilpotent.
^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. ISBN9783540426882.
^Sernesi. "An overview of classical deformation theory" (PDF). (PDF) from the original on 2020-04-27.
^Illusie, L. (PDF). Archived from the original (PDF) on 2020-11-25. Retrieved 2020-04-27.
^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. ISBN978-3-642-64766-6.
^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].
Kodaira, Kunihiko (1986), Complex manifolds and deformation of complex structures, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 283, Berlin, New York: Springer-Verlag, ISBN978-0-387-96188-0, MR 0815922
Mathoverflow post relating deformations to the jacobian ring.
April 12, 2024
kodaira, spencer, mathematics, introduced, kunihiko, kodaira, donald, spencer, associated, deformation, scheme, complex, manifold, taking, tangent, space, point, deformation, space, first, cohomology, group, sheaf, vector, fields, contents, definition, histori. 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 Contents 1 Definition 1 1 Historical motivation 1 2 Original definition 1 3 Remarks 2 Constructions 2 1 Using infinitesimals 2 1 1 Cocycle condition for deformations 2 1 2 Conversion to cocycles of vector fields 2 2 Using vector fields 2 3 In scheme theory 2 4 Of ringed topoi 3 Examples 3 1 With analytic germs 3 2 On affine hypersurfaces with the cotangent complex 4 See also 5 ReferencesDefinition editHistorical motivation edit The Kodaira Spencer map was originally constructed in the setting of complex manifolds Given a complex analytic manifold M displaystyle M nbsp with charts Ui displaystyle U i nbsp and biholomorphic maps fjk displaystyle f jk nbsp sending zk zj zj1 zjn displaystyle z k to z j z j 1 ldots z j n nbsp gluing the charts together the idea of deformation theory is to replace these transition maps fjk zk displaystyle f jk z k nbsp by parametrized transition maps fjk zk t1 tk displaystyle f jk z k t 1 ldots t k nbsp over some base B displaystyle B nbsp which could be a real manifold with coordinates t1 tk displaystyle t 1 ldots t k nbsp such that fjk zk 0 0 fjk zk displaystyle f jk z k 0 ldots 0 f jk z k nbsp This means the parameters ti displaystyle t i nbsp deform the complex structure of the original complex manifold M displaystyle M nbsp Then these functions must also satisfy a cocycle condition which gives a 1 cocycle on M displaystyle M nbsp 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 H1 M TM displaystyle H 1 M T M nbsp called the Kodaira Spencer map 1 Original definition edit More formally the Kodaira Spencer map is 2 KS T0B H1 M TM displaystyle KS T 0 B to H 1 M T M nbsp where M B displaystyle mathcal M to B nbsp is a smooth proper map between complex spaces 3 i e a deformation of the special fiber M M0 displaystyle M mathcal M 0 nbsp KS displaystyle KS nbsp is the connecting homomorphism obtained by taking a long exact cohomology sequence of the surjection TM M T0B OM displaystyle T mathcal M M to T 0 B otimes mathcal O M nbsp whose kernel is the tangent bundle TM displaystyle T M nbsp If v displaystyle v nbsp is in T0B displaystyle T 0 B nbsp then its image KS v displaystyle KS v nbsp is called the Kodaira Spencer class of v displaystyle v nbsp Remarks edit 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 k displaystyle k nbsp of characteristic 0 displaystyle 0 nbsp there is a natural bijection between isomorphisms classes of X S Spec k t t2 displaystyle mathcal X to S operatorname Spec k t t 2 nbsp and H1 X TX displaystyle H 1 X TX nbsp Constructions editUsing infinitesimals edit Cocycle condition for deformations editOver characteristic 0 displaystyle 0 nbsp 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 X displaystyle X nbsp covered by finitely many charts U Ua a I displaystyle mathcal U U alpha alpha in I nbsp with coordinates za za1 zan displaystyle z alpha z alpha 1 ldots z alpha n nbsp and transition functionsfba Ub Uab Ua Uab displaystyle f beta alpha U beta U alpha beta to U alpha U alpha beta nbsp where fab zb za displaystyle f alpha beta z beta z alpha nbsp Recall that a deformation is given by a commutative diagramX X Spec C Spec C e displaystyle begin matrix X amp to amp mathfrak X downarrow amp amp downarrow text Spec mathbb C amp to amp text Spec mathbb C varepsilon end matrix nbsp where C e displaystyle mathbb C varepsilon nbsp is the ring of dual numbers and the vertical maps are flat the deformation has the cohomological interpretation as cocycles f ab zb e displaystyle tilde f alpha beta z beta varepsilon nbsp on Ua Spec C e displaystyle U alpha times text Spec mathbb C varepsilon nbsp whereza f ab zb e fab zb ebab zb displaystyle z alpha tilde f alpha beta z beta varepsilon f alpha beta z beta varepsilon b alpha beta z beta nbsp If the f ab displaystyle tilde f alpha beta nbsp satisfy the cocycle condition then they glue to the deformation X displaystyle mathfrak X nbsp This can be read asf ag zg e f ab f bg zg e e fab fbg zg ebbg zg ebab fbg zg ebbg zg displaystyle begin aligned tilde f alpha gamma z gamma varepsilon amp tilde f alpha beta tilde f beta gamma z gamma varepsilon varepsilon amp f alpha beta f beta gamma z gamma varepsilon b beta gamma z gamma amp varepsilon b alpha beta f beta gamma z gamma varepsilon b beta gamma z gamma end aligned nbsp Using the properties of the dual numbers namely g a be g a eg a b displaystyle g a b varepsilon g a varepsilon g a b nbsp we havefab fbg zg ebbg zg fab fbg zg e fab za za bbg zg displaystyle begin aligned f alpha beta f beta gamma z gamma varepsilon b beta gamma z gamma amp f alpha beta f beta gamma z gamma varepsilon frac partial f alpha beta partial z alpha z alpha b beta gamma z gamma end aligned nbsp andebab fbg zg ebbg zg ebab fbg zg e2 bab za za bbg zg ebab fbg zg ebab zb displaystyle begin aligned varepsilon b alpha beta f beta gamma z gamma varepsilon b beta gamma z gamma amp varepsilon b alpha beta f beta gamma z gamma varepsilon 2 frac partial b alpha beta partial z alpha z alpha b beta gamma z gamma amp varepsilon b alpha beta f beta gamma z gamma amp varepsilon b alpha beta z beta end aligned nbsp hence the cocycle condition on Ua Spec C e displaystyle U alpha times text Spec mathbb C varepsilon nbsp is the following two rulesbag fab zbbbg bab displaystyle b alpha gamma frac partial f alpha beta partial z beta b beta gamma b alpha beta nbsp fag fab fbg displaystyle f alpha gamma f alpha beta circ f beta gamma nbsp Conversion to cocycles of vector fields editThe cocycle of the deformation can easily be converted to a cocycle of vector fields 8 8ab C1 U TX displaystyle theta theta alpha beta in C 1 mathcal U T X nbsp as follows given the cocycle f ab fab ebab displaystyle tilde f alpha beta f alpha beta varepsilon b alpha beta nbsp we can form the vector field8ab i 1nbabi zai displaystyle theta alpha beta sum i 1 n b alpha beta i frac partial partial z alpha i nbsp which is a 1 cochain Then the rule for the transition maps of bag displaystyle b alpha gamma nbsp gives this 1 cochain as a 1 cocycle hence a class 8 H1 X TX displaystyle theta in H 1 X T X nbsp Using vector fields editOne 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 t displaystyle t nbsp Then the cocycle condition can be read asfika zk t fija fkj1 zk t fkjn zk t t displaystyle f ik alpha z k t f ij alpha f kj 1 z k t ldots f kj n z k t t nbsp Then the derivative of fika zk t displaystyle f ik alpha z k t nbsp with respect to t displaystyle t nbsp can be calculated from the previous equation as fika zk t t fija zj t t b 0n fija zj t fjkb zk t fjkb zk t t displaystyle begin aligned frac partial f ik alpha z k t partial t amp 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 nbsp Note because zjb fjkb zk t displaystyle z j beta f jk beta z k t nbsp and zia fija zj t displaystyle z i alpha f ij alpha z j t nbsp then the derivative reads as fika zk t t fija zj t t b 0n zia zjb fjkb zk t t displaystyle begin aligned frac partial f ik alpha z k t partial t amp frac partial f ij alpha z j t partial t sum beta 0 n frac partial z i alpha partial z j beta cdot frac partial f jk beta z k t partial t end aligned nbsp With a change of coordinates of the part of the previous holomorphic vector field having these partial derivatives as the coefficients we can write zjb a 1n zia zjb zia displaystyle frac partial partial z j beta sum alpha 1 n frac partial z i alpha partial z j beta cdot frac partial partial z i alpha nbsp Hence we can write up the equation above as the following equation of vector fields a 0n fika zk t t zia a 0n fija zj t t zia b 0n fjkb zk t t zjb displaystyle begin aligned sum alpha 0 n frac partial f ik alpha z k t partial t frac partial partial z i alpha amp sum alpha 0 n frac partial f ij alpha z j t partial t frac partial partial z i alpha amp sum beta 0 n frac partial f jk beta z k t partial t frac partial partial z j beta end aligned nbsp Rewriting this as the vector fields8ik t 8ij t 8jk t displaystyle theta ik t theta ij t theta jk t nbsp where8ij t fija zj t t zia displaystyle theta ij t frac partial f ij alpha z j t partial t frac partial partial z i alpha nbsp gives the cocycle condition Hence 8ij displaystyle theta ij nbsp has an associated class in 8ij H1 M TM displaystyle theta ij in H 1 M T M nbsp from the original deformation f ij displaystyle tilde f ij nbsp of fij displaystyle f ij nbsp In scheme theory editDeformations of a smooth variety 5 X X Spec k Spec k e displaystyle begin matrix X amp to amp mathfrak X downarrow amp amp downarrow text Spec k amp to amp text Spec k varepsilon end matrix nbsp have a Kodaira Spencer class constructed cohomologically Associated to this deformation is the short exact sequence0 p WSpec k e 1 WX1 WX S1 0 displaystyle 0 to pi Omega text Spec k varepsilon 1 to Omega mathfrak X 1 to Omega mathfrak X S 1 to 0 nbsp where p X Spec k e displaystyle pi mathfrak X to text Spec k varepsilon nbsp which when tensored by the OX displaystyle mathcal O mathfrak X nbsp module OX displaystyle mathcal O X nbsp gives the short exact sequence0 OX WX1 OX WX1 0 displaystyle 0 to mathcal O X to Omega mathfrak X 1 otimes mathcal O X to Omega X 1 to 0 nbsp Using derived categories this defines an element inRHom WX1 OX 1 RHom OX TX 1 Ext1 OX TX H1 X TX displaystyle begin aligned mathbf R text Hom Omega X 1 mathcal O X 1 amp cong mathbf R text Hom mathcal O X T X 1 amp cong text Ext 1 mathcal O X T X amp cong H 1 X T X end aligned nbsp generalizing the Kodaira Spencer map Notice this could be generalized to any smooth map f X Y displaystyle f X to Y nbsp in Sch S displaystyle text Sch S nbsp using the cotangent sequence giving an element in H1 X TX Y f WY Z1 displaystyle H 1 X T X Y otimes f Omega Y Z 1 nbsp Of ringed topoi editOne of the most abstract constructions of the Kodaira Spencer maps comes from the cotangent complexes associated to a composition of maps of ringed topoiX fY Z displaystyle X xrightarrow f Y to Z nbsp Then associated to this composition is a distinguished trianglef LY Z LX Z LX Y 1 displaystyle f mathbf L Y Z to mathbf L X Z to mathbf L X Y xrightarrow 1 nbsp and this boundary map forms the Kodaira Spencer map 6 or cohomology class denoted K X Y Z displaystyle K X Y Z nbsp If the two maps in the composition are smooth maps of schemes then this class coincides with the class in H1 X TX Y f WY Z1 displaystyle H 1 X T X Y otimes f Omega Y Z 1 nbsp Examples editWith analytic germs editThe 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 f z1 zn C z1 zn H displaystyle f z 1 ldots z n in mathbb C z 1 ldots z n H nbsp its space of deformations can be given by the moduleT1 Hdf Hn displaystyle T 1 frac H df cdot H n nbsp For example if f y2 x3 displaystyle f y 2 x 3 nbsp then its versal deformations is given byT1 C x y y x2 displaystyle T 1 frac mathbb C x y y x 2 nbsp hence an arbitrary deformation is given by F x y a1 a2 y2 x3 a1 a2x displaystyle F x y a 1 a 2 y 2 x 3 a 1 a 2 x nbsp Then for a vector v T0 C2 displaystyle v in T 0 mathbb C 2 nbsp which has the basis a1 a2 displaystyle frac partial partial a 1 frac partial partial a 2 nbsp there the map KS v v F displaystyle KS v mapsto v F nbsp sendingϕ1 a1 ϕ2 a2 ϕ1 F a1 ϕ2 F a2 ϕ1 ϕ2 x displaystyle begin aligned phi 1 frac partial partial a 1 phi 2 frac partial partial a 2 mapsto amp phi 1 frac partial F partial a 1 phi 2 frac partial F partial a 2 amp phi 1 phi 2 cdot x end aligned nbsp On affine hypersurfaces with the cotangent complex editFor an affine hypersurface i X0 An Spec k displaystyle i X 0 hookrightarrow mathbb A n to text Spec k nbsp over a field k displaystyle k nbsp defined by a polynomial f displaystyle f nbsp there is the associated fundamental trianglei LAn Spec k LX0 Spec k LX0 An 1 displaystyle i mathbf L mathbb A n text Spec k to mathbf L X 0 text Spec k to mathbf L X 0 mathbb A n xrightarrow 1 nbsp Then applying RHom OX0 displaystyle mathbf RHom mathcal O X 0 nbsp gives the long exact sequenceRHom i LAn Spec k OX0 1 RHom LX0 Spec k OX0 1 RHom LX0 An OX0 1 RHom i LAn Spec k OX0 RHom LX0 Spec k OX0 RHom LX0 An OX0 displaystyle begin aligned amp textbf RHom i mathbf L mathbb A n text Spec k mathcal O X 0 1 leftarrow textbf RHom mathbf L X 0 text Spec k mathcal O X 0 1 leftarrow textbf RHom mathbf L X 0 mathbb A n mathcal O X 0 1 leftarrow amp textbf RHom i mathbf L mathbb A n text Spec k mathcal O X 0 leftarrow textbf RHom mathbf L X 0 text Spec k mathcal O X 0 leftarrow textbf RHom mathbf L X 0 mathbb A n mathcal O X 0 end aligned nbsp Recall that there is the isomorphismRHom LX0 Spec k OX0 1 Ext1 LX0 Spec k OX0 displaystyle textbf RHom mathbf L X 0 text Spec k mathcal O X 0 1 cong text Ext 1 mathbf L X 0 text Spec k mathcal O X 0 nbsp 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 LAn Spec k WAn Spec k 1 displaystyle mathbf L mathbb A n text Spec k cong Omega mathbb A n text Spec k 1 nbsp is a free module RHom i LAn Spec k OX0 1 0 displaystyle textbf RHom i mathbf L mathbb A n text Spec k mathcal O X 0 1 0 nbsp Also because LX0 An I I2 1 displaystyle mathbf L X 0 mathbb A n cong mathcal I mathcal I 2 1 nbsp there are isomorphismsRHom LX0 An OX0 1 RHom I I2 1 OX0 1 RHom I I2 OX0 Ext0 I I2 OX0 Hom I I2 OX0 OX0 displaystyle begin aligned textbf RHom mathbf L X 0 mathbb A n mathcal O X 0 1 cong amp textbf RHom mathcal I mathcal I 2 1 mathcal O X 0 1 cong amp textbf RHom mathcal I mathcal I 2 mathcal O X 0 cong amp text Ext 0 mathcal I mathcal I 2 mathcal O X 0 cong amp text Hom mathcal I mathcal I 2 mathcal O X 0 cong amp mathcal O X 0 end aligned nbsp The last isomorphism comes from the isomorphism I I2 I OAnOX0 displaystyle mathcal I mathcal I 2 cong mathcal I otimes mathcal O mathbb A n mathcal O X 0 nbsp and a morphism inHomOX0 I OAnOX0 OX0 displaystyle text Hom mathcal O X 0 mathcal I otimes mathcal O mathbb A n mathcal O X 0 mathcal O X 0 nbsp send gf g g f displaystyle gf mapsto g g f nbsp giving the desired isomorphism From the cotangent sequence f f 2 g dg 1WAn1 OX0 WX0 Spec k 1 0 displaystyle frac f f 2 xrightarrow g mapsto dg otimes 1 Omega mathbb A n 1 otimes mathcal O X 0 to Omega X 0 text Spec k 1 to 0 nbsp which is a truncated version of the fundamental triangle the connecting map of the long exact sequence is the dual of g dg 1 displaystyle g mapsto dg otimes 1 nbsp giving the isomorphismExt1 LX0 k OX0 k x1 xn f f x1 f xn displaystyle text Ext 1 mathbf L X 0 k mathcal O X 0 cong frac k x 1 ldots x n left f frac partial f partial x 1 ldots frac partial f partial x n right nbsp Note this computation can be done by using the cotangent sequence and computing Ext1 WX01 OX0 displaystyle text Ext 1 Omega X 0 1 mathcal O X 0 nbsp 8 Then the Kodaira Spencer map sends a deformationk e x1 xn f eg displaystyle frac k varepsilon x 1 ldots x n f varepsilon g nbsp to the element g Ext1 LX0 k OX0 displaystyle g in text Ext 1 mathbf L X 0 k mathcal O X 0 nbsp See also editDeformation theory Cotangent complex Schlessinger s theorem characteristic linear system of an algebraic family of curves Gauss Manin connection Derived category Ext functorReferences edit 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 Huybrechts 2005 6 2 6 The main difference between a complex manifold and a complex space is that the latter is allowed to have a nilpotent 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 Sernesi An overview of classical deformation theory PDF Archived PDF from the original on 2020 04 27 Illusie L Complexe cotangent application a la theorie des deformations PDF Archived from the original PDF on 2020 11 25 Retrieved 2020 04 27 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 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 Huybrechts Daniel 2005 Complex Geometry An Introduction Springer ISBN 3 540 21290 6 Kodaira Kunihiko 1986 Complex manifolds and deformation of complex structures Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences vol 283 Berlin New York Springer Verlag ISBN 978 0 387 96188 0 MR 0815922 Mathoverflow post relating deformations to the jacobian ring Retrieved from https en wikipedia org w index php title Kodaira Spencer map amp oldid 1068885821, wikipedia, wiki, book, books, library,