View source for Deformation

A deformation of an analytic structure is a family of analytic spaces (or analytic objects connected with these spaces) depending on parameters (cf. [[Analytic space|Analytic space]]). The theory of deformations originated with the problem of classification of all possible pairwise non-isomorphic complex structures on a given differentiable real manifold. The fundamental idea (which must be credited to B. Riemann) was to introduce an analytic structure on the set of all such structures. The idea was made precise by the following concepts. An analytic family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307001.png" /> of complex manifolds parametrized by a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307002.png" /> is defined as any smooth (i.e. locally structured as a projection of a direct product with smooth factors) analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307003.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307004.png" /> is connected, then all fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307005.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307006.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307007.png" /> are diffeomorphic to a fixed fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307008.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d0307009.png" />, and may be considered as a family of complex structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070010.png" />, analytically dependent on the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070011.png" />. If the fibres of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070012.png" /> consist all of complex manifolds diffeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070013.png" /> and if all fibres are pairwise non-isomorphic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070014.png" /> is said to be the moduli space of the real manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070015.png" />. A moduli space can also be defined for manifolds belonging to a specified class. The problem of constructing a moduli space (or the moduli problem) was first solved for compact Riemann surfaces (cf. [[Riemann surfaces, conformal classes of|Riemann surfaces, conformal classes of]]). Similar, though incomplete, results were also obtained for compact manifolds of complex dimension 2 (cf. [[Analytic surface|Analytic surface]]).

Considerable difficulties are encountered in the study of moduli problems for higher-dimensional manifolds. In this context K. Kodaira and D.C. Spencer [[#References|[6]]], [[#References|[7]]], [[#References|[8]]] made a local study of moduli problems, thus laying the foundations of the theory of deformation of complex manifolds and analytic bundles. An analytic deformation of a complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070016.png" /> is an analytic family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070018.png" /> is a [[Complex space|complex space]] with a marked point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070019.png" />, with the fibre over it coinciding with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070020.png" />. The deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070021.png" /> is said to be trivial. A deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070022.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070023.png" /> is said to be isomorphic to the deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070024.png" /> if there exists an analytic isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070025.png" /> which is the identity on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070026.png" /> and which is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070027.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070028.png" /> is an analytic deformation, then any analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070030.png" /> is a space with a marked point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070032.png" />, defines by a [[Base change|base change]] a deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070033.png" />; the inverse image of this deformation under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070034.png" />. The deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070035.png" /> is said to be locally complete (at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070036.png" />) if any analytic deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070037.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070038.png" /> is isomorphic in some neighbourhood of the marked point to its inverse image for some local analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070039.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070040.png" /> is unambiguously defined, the deformation is said to be versal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070041.png" />, and if the germ of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070042.png" /> is uniquely defined, the deformation is said to be universal. An important role in the theory is played by the linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070043.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070044.png" /> is the sheaf of germs of holomorphic vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070045.png" />, which is associated an analytic deformation and is named the corresponding infinitesimal deformation.

The principal theorem of the local theory of deformations, proved by M. Kuranishi [[#References|[9]]], states that for each compact complex manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070046.png" /> there exists a deformation versal at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070047.png" />, which is parametrized by a (not necessarily smooth) analytic subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070048.png" /> in a neighbourhood of zero of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070049.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070050.png" /> is the fibre at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070051.png" /> of some local analytic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070052.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070054.png" /> is the operation in the graded Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070055.png" /> induced by the Lie bracket in the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070056.png" />, the dots denoting terms of order 3 or higher. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070057.png" />, the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070058.png" /> is rigid, i.e. any deformation of it is locally trivial (the Fröhlicher–Nijenhuis rigidity theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070060.png" /> is a neighbourhood of zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070061.png" />. The tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070062.png" /> always coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070063.png" />. A deformation is complete at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070064.png" /> if and only if the corresponding infinitesimal deformation is surjective, and versality is equivalent to bijectivity of the infinitesimal deformation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070066.png" />, is constant in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070067.png" />, the Kuranishi deformation is universal.

The local theory of deformations of compact complex manifolds may be generalized to include the case of compact complex spaces. The requirements that the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070068.png" /> be smooth and that the fibres be compact are then replaced by the requirements that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070069.png" /> be a proper flat mapping. Here, too, it is possible to prove the existence of a deformation versal at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070070.png" /> [[#References|[3]]], [[#References|[5]]], [[#References|[11]]].

Studies are also conducted on the deformation of germs of analytic spaces (or, which amounts to the same thing, of analytic algebras). The theorem on the existence of a versal deformation for an isolated singular point of a complex space is valid [[#References|[4]]].

In addition to the deformation theories of complex spaces, there also exist deformation theories of various  "analytic objects" : analytic bundles, subspaces, mappings, cohomology classes, analytic spaces with additional structures (e.g. with polarization), etc. The principal deformations and the problems involved in these theories are similar to the ones described above. The results obtained for principal analytic bundles are also analogous to the ones above. In particular, for any [[Principal analytic fibration|principal analytic fibration]] (bundle) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070071.png" /> with compact base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070072.png" /> and complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070073.png" /> as structure group there exists a deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070074.png" />, versal at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070075.png" />, parametrized by an analytic subspace in a neighbourhood of zero of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070076.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070077.png" /> is the sheaf of germs of holomorphic sections of the vector bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070078.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070079.png" /> by the adjoint representation [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070080.png" /> is a compact Riemann surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070081.png" /> is a reductive algebraic group, it is possible to construct moduli spaces for stable principal analytic bundles. In the theory of deformation of subspaces, on the contrary, one obtains quite general results of a global nature. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070082.png" /> is an arbitrary complex space of finite dimension, a flat analytic family of compact analytic subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070083.png" /> (i.e. an analytic subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070084.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070085.png" /> is a complex space and the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070086.png" /> is a proper flat mapping) has been constructed [[#References|[2]]], and is a universal (in the global sense) deformation for any compact analytic subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070087.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070088.png" /> is the moduli space for this problem. A similar moduli problem has also been solved in a related case, and also for compact analytic cycles of a given complex space. The solution of the moduli problem for compact subspaces also entails the solution of the moduli problem for analytic mappings of a given compact complex space into another given complex space.

Attempts have been made at a unification of the above deformation theories. Each one of these theories may be related to a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070089.png" /> from the category of analytic spaces (or germs of analytic spaces) into the category of sets. For instance, in the theory of local deformations of a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070090.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070091.png" /> consists of classes of locally isomorphic deformations of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070092.png" /> parametrized by an analytic space germ <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070093.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070094.png" /> and an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070095.png" /> are fixed, there results a morphism of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070096.png" />. The surjectivity of this morphism (the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070097.png" /> is said to be complete in such a case) corresponds to the property of completeness of the deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070098.png" />, while the bijectivity corresponds to the property of its universality. In this way the moduli problem is connected with the problem of representability of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d03070099.png" />. This stimulated the study of covariant functors from the category of Artinian rings into the category of sets satisfying certain natural conditions [[#References|[12]]]. The existence of a complete pair can be proved, but only in the category of formal algebras, which corresponds to the existence of a formally complete deformation (cf. Deformation of an algebraic variety below).

A generalization of the theory of deformations of complex structures on manifolds is the theory of deformations of a [[Pseudo-group structure|pseudo-group structure]], the subject of which are families of pseudo-group structures smoothly depending on a parameter which assumes values in a real-analytic space. In particular, the existence of a versal deformation germ has been proved for a pseudo-group structure on a compact smooth manifold, corresponding to an elliptic transitive pseudo-group of transformations [[#References|[10]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.F. Donin,   "Construction of a versal family of deformations for holomorphic bundles over a compact complex space"  ''Math. USSR Sb.'' , '''23''' :  3  (1974)  pp. 405–416  ''Mat. Sb.'' , '''94''' :  3  (1974)  pp. 430–443  {{MR|393581}} {{ZBL|0325.32008}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Douady,   "Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné"  ''Ann. Inst. Fourier'' , '''16'''  (1966)  pp. 1–95  {{MR|0210935}} {{MR|0203082}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Douady,   "Le problème des modules locaux pour les espaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700100.png" />-analytiques compacts"  ''Ann. Sci. Ecole Norm. Sup.'' , '''7''' :  4  (1974)  pp. 569–602  {{MR|382729}} {{ZBL|0282.32014}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Grauert,   "Ueber die Deformation isolierter Singularitäten analytischer Menge"  ''Invent. Math.'' , '''15''' :  3  (1972)  pp. 171–198    {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H. Grauert,   "Der Satz von Kuranishi für kompakte komplexe Räume"  ''Invent. Math.'' , '''25''' :  2  (1974)  pp. 107–142  {{MR|0346194}} {{ZBL|0286.32015}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Kodaira,   D.C. Spencer,   "On deformations of complex-analytic structures I"  ''Ann. of Math.'' , '''67''' :  2  (1958)  pp. 328–401  {{MR|0112154}} {{ZBL|0128.16901}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  K. Kodaira,   D.C. Spencer,   "On deformations of complex-analytic structures II"  ''Ann. of Math.'' , '''67''' :  3  (1958)  pp. 403–466  {{MR|112154}} {{ZBL|0128.16901}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  K. Kodaira,   D.C. Spencer,   "On deformations of complex-analytic structures III"  ''Ann. of Math.'' , '''71''' :  1  (1960)  pp. 43–76  {{MR|0115189}} {{ZBL|0128.16902}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M. Kuranishi,   "New proof for the existence of locally complete families of complex structures" , ''Proc. Conf. Complex Analysis (Minneapolis, 1964)'' , Springer  (1965)  pp. 142–154  {{MR|0176496}} {{ZBL|0144.21102}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  S.H. Moolgavkar,   "On the existence of a universal germ of deformations for elliptic pseudogroup structures on compact manifolds"  ''Trans. Amer. Math. Soc.'' , '''212''' :  485  (1975)  pp. 173–197  {{MR|0458515}} {{ZBL|0308.58014}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  V.P. Palamodov,   "Deformations of complex spaces"  ''Russian Math. Surveys'' , '''31''' :  3  (1976)  pp. 129–197  ''Usphekhi Mat. Nauk'' , '''31'''  (1976)  pp. 129–194  {{MR|0508121}} {{MR|0435451}} {{ZBL|0355.32020}} {{ZBL|0347.32009}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  M. Schlessinger,   "Functors of Artin rings"  ''Trans. Amer. Math. Soc.'' , '''130'''  (1968)  pp. 208–222  {{MR|0217093}} {{ZBL|0167.49503}} </TD></TR></table>



====Comments====
Instead of  "infinitesimal deformation"  it is usual to speak of  "Kodaira–Spencer mappingKodaira–Spencer mapping" . For some results and concepts cf. [[#References|[a1]]]–[[#References|[a5]]]. A comprehensive treatment of the Kodaira–Spencer theory of deformation is [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Gasqui,   H. Goldschmidt,   "Complexes of differential operators and symmetric spaces"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 797–828  {{MR|0981631}} {{ZBL|0666.58041}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Gasqui,   H. Goldschmidt,   "Some rigidity results in the deformation theory of symmetric spaces"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 839–851  {{MR|0981633}} {{ZBL|0698.53034}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Gasqui,   H. Goldschmidt,   "Deformations infinitesimals des structures conformes plates" , Birkhäuser  (1984)    {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Hermann,   "Geometric and Lie-theoretic principles in pure and applied deformation theory"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 701–796  {{MR|0981630}} {{ZBL|0685.58038}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.F. Pommaret,   "Deformation theory of geometric and algebraic structures"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 829–838  {{MR|0981632}} {{ZBL|0684.58047}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Kodaira,   "Complex manifolds and deformations of complex structures" , Springer  (1986)  {{MR|0815922}} {{ZBL|}} </TD></TR></table>

A deformation of an algebraic variety is the inclusion of the algebraic variety into a family of algebraic varieties. The theory of deformation of algebraic varieties and schemes is the algebraic analogue of the theory of deformation of analytic structures. Its principal problems are listed below.

Existence of a lift. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700101.png" /> be a scheme over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700102.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700103.png" /> be a scheme and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700104.png" /> be a point with the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700105.png" />. Does a flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700106.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700107.png" /> for which the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700108.png" /> over the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700109.png" /> is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700110.png" /> exist? (The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700111.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700112.png" /> is known as the deformation or lift of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700113.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700114.png" />.)

The universality problem. Does a versal (respectively, universal) deformation of a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700115.png" /> (i.e. a deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700116.png" /> over the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700117.png" /> such that for any other deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700118.png" /> it is possible to find a (respectively, a unique) morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700119.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700120.png" />) exist?

Any deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700121.png" /> of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700122.png" /> by the operation of formal completion along the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700123.png" /> defines a formal deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700124.png" /> over the completion of the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700125.png" /> of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700126.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700127.png" />, i.e. a dense formal scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700128.png" /> with topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700129.png" />. The formal analogues of the problems listed above are formulated as follows:

Existence of a formal deformation. Let a complete local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700130.png" /> with the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700131.png" /> be given. Does a flat formal scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700132.png" /> with topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700133.png" /> exist?

Existence of a formal moduli scheme. Does a formal versal (respectively, universal) deformation, i.e. a flat formal scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700134.png" /> over a complete local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700135.png" /> with the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700136.png" /> such that for any formal deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700137.png" /> there is a (respectively, a unique) ring homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700138.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700139.png" />, exist?

A universal formal deformation of a smooth variety represents the algebraic analogue of a local moduli space in the theory of deformation of analytic structures.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700140.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700141.png" /> is a local Artinian (respectively, complete) ring with residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700142.png" />, the deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700143.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700144.png" /> is said to be infinitesimal (respectively, effective formal). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700145.png" /> is a complete local ring of characteristic zero (e.g. the ring of Witt vectors, cf. [[Witt vector|Witt vector]]), the effective formal deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700146.png" /> is said to be the lift of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700147.png" /> into characteristic zero.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700148.png" /> is a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700149.png" />-scheme and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700150.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700151.png" /> is the tangent bundle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700152.png" />, then for any Artinian (respectively, complete) local ring there exists an infinitesimal (respectively, formal) deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700153.png" />. Moreover, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700154.png" />, such a deformation is unique up to isomorphism [[#References|[4]]]. Similar statements for not necessarily smooth schemes are given in terms of the cotangent complex [[#References|[5]]], [[#References|[6]]]. The problem of the existence of an effective formal deformation is studied by means of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700155.png" /> from the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700156.png" /> of local Artinian rings with residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700157.png" /> into the category of sets which associates to each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700158.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700159.png" /> the set of all infinitesimal deformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700160.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700161.png" />. A universal formal deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700162.png" /> exists if and only if the functor is a pro-representable functor. Here, a pro-representing object — a complete local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700163.png" /> with residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700164.png" /> — is said to be a formal scheme of moduli of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700165.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700166.png" />. A formal versal deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700167.png" /> exists if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700168.png" /> is proper over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700169.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700170.png" /> is an affine scheme of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700171.png" /> with isolated singular points [[#References|[2]]], [[#References|[6]]]. A versal formal deformation is universal if for any surjective homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700172.png" /> of Artinian local rings, and any deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700173.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700174.png" /> the natural mapping of automorphism groups

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700175.png" /></td> </tr></table>

is surjective. This condition is satisfied, for example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700176.png" /> is a smooth scheme and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700177.png" />. Here, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700178.png" />, the formal scheme of moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700179.png" /> is a complete regular local ring which is isomorphic to the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700180.png" /> of formal power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700181.png" /> variables. In this case the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700182.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700183.png" /> and is called the number of local moduli of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700184.png" />. In the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700185.png" /> is equal to the dimension of the space tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700186.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700187.png" />, i.e. to the dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700188.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700189.png" /> is the maximal ideal of the corresponding local ring, and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700190.png" /></td> </tr></table>

The presence of nilpotent elements in a formal scheme of moduli is a fairly frequent phenomenon.

If the versal (respectively, universal) formal deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700191.png" /> is algebraizable, i.e. if there exists a flat scheme over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700192.png" /> the formal completion of which along a closed fibre is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700193.png" />, the corresponding algebraization is said to be a local versal (respectively, universal) deformation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700194.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700195.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700196.png" /> is projective and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700197.png" />, the algebraization exists. Thus, for a smooth curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700198.png" /> there exists a local universal deformation over the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700199.png" />. In the general case, for each polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700200.png" /> of the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700201.png" /> there exists a maximal closed subscheme of moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700202.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700203.png" /> is algebraizable. The codimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700204.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700205.png" /> does not exceed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700206.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700207.png" /> is an algebraic [[K3-surface|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700208.png" />-surface]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700209.png" /> is regular of dimension 20, and for any polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700210.png" /> the subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700211.png" /> is regular of dimension 19.

Artin's approximation theorem is used in the algebraization of formal schemes of moduli. There exists a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700212.png" /> of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700213.png" /> and a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700214.png" /> with residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700215.png" /> such that the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700216.png" />, and there exists a deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700217.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700218.png" /> inducing the versal local deformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700219.png" />. The scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700220.png" /> is defined uniquely up to a local isomorphism in the étale topology [[#References|[1]]]. For the deformation of singular varieties and singular points see [[Singular point|Singular point]] of an algebraic variety. For the deformation of group schemes see [[Group scheme|Group scheme]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Artin,   "Algebraic approximation of structures over complete local rings"  ''Publ. Math. IHES'' , '''36'''  (1969)  pp. 23–58  {{MR|0268188}} {{ZBL|0181.48802}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.V. Dolgachev,   V.A. Iskovskikh,   "Geometry of algebraic varieties"  ''Russian Math. Surveys'' , '''5''' :  6  (1976)  pp. 803–864  ''Itogi Nauk. i Tekhn. Algebra Topol. Geom.'' , '''12'''  (1974)  pp. 77–170    {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Schlessinger,   "Functors of Artin rings"  ''Trans. Amer. Math. Soc.'' , '''130'''  (1968)  pp. 208–222  {{MR|0217093}} {{ZBL|0167.49503}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Grothendieck (ed.)  et al. (ed.) , ''Révètement étales et groupe fondamental, SGA 1'' , ''Lect. notes in math.'' , '''224''' , Springer  (1971)  {{MR|2017446}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  D.S. Rim,   "Formal deformation theory"  A. Grothendieck (ed.)  et al. (ed.) , ''Groupes de monodromie en géométrie algébrique. SGA 7'' , ''Lect. notes in math.'' , '''288''' , Springer  (1972)  pp. 32–132  {{MR|}} {{ZBL|0246.14001}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D. Mumford,   O. Zariski (ed.) , ''Algebraic surfaces'' , Springer  (1971)  pp. 118–128  {{MR|0469915}} {{ZBL|0138.42002}} </TD></TR></table>

''I.V. Dolgachev''

A deformation of algebras is a family of algebras depending or parameters. All possible bilinear operations or algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700221.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700222.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700223.png" /> form a vector space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700224.png" />. Two elements of this space represent isomorphic algebras if and only if they lie on the same orbit of the linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700225.png" />, acting naturally on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700226.png" />. The theory of deformation of algebras makes it possible to study the local structure of the quotient set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700227.png" />, i.e. the set of classes of isomorphic algebras in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700228.png" />, a direct description of which involves considerable difficulties. If a certain class of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700229.png" /> is isolated, one may consider the deformation of the algebras in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700230.png" /> that stay within that class. In particular, one may consider deformations of associative and associative-commutative algebras and Lie algebras, which form the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700231.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700232.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700233.png" />, respectively, which are invariant with respect to the action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700234.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700235.png" />, these classes are algebraic varieties in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700236.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700237.png" />.

The theory of deformation of algebras in a finite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700238.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700239.png" /> of real or complex numbers in many respects resembles the theory of deformations of analytic structures. Each finite-dimensional algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700240.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700241.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700242.png" /> has a complete deformation parametrized by an analytic subspace germ in the zero of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700243.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700244.png" /> this subspace coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700245.png" />). The rigidity theorem immediately follows from this fundamental result: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700246.png" />, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700247.png" /> is rigid in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700248.png" />, i.e. the orbit of an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700249.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700250.png" /> is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700251.png" />. Thus, semi-simple Lie algebras, as well as their Borel subalgebras, are rigid in the class of Lie algebras. The converse proposition is not true. Analogous theorems are valid for finite-dimensional algebras over an arbitrary algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700252.png" />. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700253.png" />, the orbit of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700254.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700255.png" /> is Zariski open.

The theory of deformation of homomorphisms of one finite-dimensional algebra into another may be developed in an analogous manner. In fact, the theories described above form part of a general scheme which involves the use of graded Lie algebras. Similar results have also been obtained for the deformation of subalgebras.

In addition to the above theory there also exists the theory of formal deformations of algebras and their homomorphisms over an arbitrary field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700256.png" />. A formal deformation of an algebra defined in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700257.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700258.png" /> is an algebra over the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700259.png" /> over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700260.png" /> of formal power series over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700261.png" /> with the operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700262.png" /> which is fully determined by the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700263.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700264.png" />. In requiring the deformation to belong to a given class of algebras, one may speak of formal deformations of associative, associative-commutative, Lie, and other algebras.

Two formal deformations of an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700265.png" /> with the multiplications <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700266.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700267.png" />, respectively, are said to be equivalent if there exists a linear automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700268.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700269.png" /> with the property

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700270.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700271.png" /> is a linear mapping, such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700272.png" /></td> </tr></table>

A deformation which is equivalent to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700273.png" /> with the initial multiplication is known as trivial. An algebra without non-trivial formal deformations in a given class of algebras is known as formally rigid in that class. For instance, the free algebra in the class of all algebras will be formally rigid. In the classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700274.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700275.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700276.png" /> the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700277.png" /> is a sufficient condition of formal rigidity of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700278.png" />; in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700279.png" /> it is also a necessary condition [[#References|[3]]].

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700280.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700281.png" />, formal deformations of algebras are a tool in the study of analytic deformations.

An important field of applications and a source of examples of deformations of algebras is theoretical physics, in which, in particular, the following class of deformations of an algebra arose [[#References|[4]]], [[#References|[5]]]. A contraction of a finite-dimensional algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700282.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700283.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700284.png" /> is a continuous curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700285.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700286.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700287.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700288.png" /> by a contraction is known as the limit algebra and need not be isomorphic to the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700289.png" />. For instance, any algebra may be contracted into an algebra with zero multiplication; any semi-simple Lie algebra may be contracted into a non-Abelian non-semi-simple algebra.

====References====
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  A. Nijenhuis,   R.W. Richardson Jr.,   "Cohomology and deformations in graded Lie algebras"  ''Bull. Amer. Math. Soc.'' , '''72''' :  1  (1966)  pp. 1–29  {{MR|0195995}} {{ZBL|0136.30502}} </TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  A. Nijenhuis,   R.W. Richardson Jr.,   "Deformations of homomorphisms of Lie groups and Lie algebras"  ''Bull. Amer. Math. Soc.'' , '''73''' :  1  (1967)  pp. 175–179  {{MR|0204575}} {{ZBL|0153.04402}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Gerstenhaber,   "On the deformation of rings and algebras"  ''Ann. of Math.'' , '''79'''  (1964)  pp. 59–103  {{MR|0171807}} {{ZBL|0123.03101}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  D. Knudson,   "On the deformation of commutative algebras"  ''Trans. Amer. Math. Soc.'' , '''140'''  (1969)  pp. 55–70  {{MR|0240091}} {{ZBL|0199.36102}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Inönü,   E.P. Wigner,   "On the contraction of groups and their representations"  ''Proc. Nat. Acad. Sci. USA'' , '''39''' :  6  (1953)  pp. 510–524  {{MR|0055352}} {{ZBL|0050.02601}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R. Hermann,   "Lie groups for physicists" , Benjamin  (1966)  {{MR|0213463}} {{ZBL|0135.06901}} </TD></TR></table>

''A.A. BoyarkinA.V. MikhalevA.L. Onishchik''

====Comments====
In recent years ideas from [[Non-standard analysis|non-standard analysis]] have been applied to deformation problems in algebra. In particular, by this approach rigidity results for Lie algebras have been obtained also in cases with non-vanishing cohomology. For the use of techniques from non-standard analysis in the theory of deformation of algebras, cf. [[#References|[a9]]], [[#References|[a10]]].

It has now become clear that the deformation theory of algebras is not merely analogous to that of analytic structures but that there are in fact fundamental relations between the two subjects. In particular, the formal deformation theory of a smooth compact complex algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700290.png" /> can be reduced to that of a single ring built from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700291.png" />. There is also a relationship between the classical Hodge decomposition of the cohomology of an analytic manifold and the recent Hodge decomposition of the cohomology of a commutative algebra, [[#References|[a1]]].

For the relation between deformations of singular and non-singular Riemann surfaces and the deformation theory of the corresponding algebra of analytic functions, cf. [[#References|[a2]]].

In the theory of operator and Banach algebras one studies whether algebras close to the given one (in the metric sense) are isomorphic, cf. [[#References|[a3]]]–[[#References|[a5]]].

There is a deformation-theoretic approach to quantization based on the so-called star-products, which are non-commutative deformations of the commutative product of the ring of functions on a manifold, cf. [[#References|[a6]]]–[[#References|[a8]]].

For deformations of meromorphic differential equations cf. [[#References|[a11]]].

Reference [[#References|[a12]]] provides an up-to-date (1988) survey of many aspects of deformation theory, with the notable omission of deformations (bendings) of convex surfaces, resolution of singularities, deformations of vector fields and such in the setting of structural stability of dynamical systems, isospectral and isomonodromy deformations in the setting of integrable systems, and deformations of mechanical structures and solid bodies. Cf. also [[Deformation, isometric|Deformation, isometric]]; [[Deformation over a principal base|Deformation over a principal base]]; [[Deformation tensor|Deformation tensor]]; [[Integrable system|Integrable system]]; [[Resolution of singularities|Resolution of singularities]]; [[Stability of an elastic system|Stability of an elastic system]]; [[Stability in the presence of persistently acting perturbations|Stability in the presence of persistently-acting perturbations]]; and [[Perturbation theory|Perturbation theory]], which contain information on other aspects of deformation theory than the ones dealt with above and in [[#References|[a12]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Gerstenhaber,   D. Shack,   "Algebraic cohomology and deformation theory"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 11–264  {{MR|0981619}} {{ZBL|0676.16022}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Rochberg,   "Deformation theory for algebras of analytic functions"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 501–536  {{MR|0981625}} {{ZBL|0701.46042}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E. Christensen,   "Close operator algebras"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 537–556  {{MR|0981626}} {{ZBL|0672.46031}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Jarosz,   "Perturbations of Banach algebras" , Springer  (1985)  {{MR|0788884}} {{ZBL|0557.46029}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.E. Johnson,   "Perturbations of multiplication and homomorphisms"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 565–579  {{MR|0981628}} {{ZBL|0676.46051}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Lichnerowicz,   "Applications of the deformations of algebraic structures to geometry and mathematical physics"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 855–896  {{MR|981634}} {{ZBL|0674.58021}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  F. Bayen,   M. Flato,   C. Fronsdal,   A. Lichnerowicz,   D. Sternheimer,   "Deformation theory and quantization"  ''Ann. of Physics'' , '''111'''  (1978)  pp. 61–110; 111–151  {{MR|0496158}} {{MR|0496157}} {{ZBL|0377.53025}} {{ZBL|0377.53024}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  M. de Wilde,   P. Lecomte,   "Invariant deformations of the Poisson Lie algebra of a symplectic manifold and star-products"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 897–960    {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Goze,   "Perturbations of Lie algebra structures"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 265–356  {{MR|0981620}} {{ZBL|0715.17009}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J.M. Ancochea Bermudez,   "On the rigidity of solvable Lie algebras"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 403–446  {{MR|}} {{ZBL|0672.17006}} </TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  D.G. Babbitt,   V.S. Varadarajan,   "Local isoformal deformation theory for meromorphic differential equations near an irregular singularity"  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  pp. 583–700  {{MR|0981629}} {{ZBL|0668.34010}} </TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)  {{MR|0981617}} {{ZBL|0654.00006}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  D. Sternheimer,   "Star products: their ubiquity and unicity"  J. Bertrand (ed.)  et al. (ed.) , ''Modern Group Theoretical Methods in Physics: Proc. 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A deformation of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700292.png" /> of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700293.png" /> is a [[Homotopy|homotopy]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700294.png" /></td> </tr></table>

for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700295.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700296.png" />. If, in addition, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700297.png" /> belongs to some subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700298.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700299.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700300.png" /> is said to be a deformation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700301.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700302.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700303.png" /> is said to be deformable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700304.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700305.png" />. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700306.png" /> is said to be deformable into a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700307.png" /> if it is deformable in itself in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700308.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700309.png" /> is contractible if and only if it is deformable in one of its points. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700310.png" /> is deformable into a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700311.png" /> if and only if for the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700312.png" /> the right homotopy inverse mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700313.png" /> exists, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030700/d030700314.png" />. The concept of deformation of any space in itself in a subspace is related to the concept of a weak [[Retraction|retraction]].

''M.I. Voitsekhovskii''