# User:Maximilian Janisch/latexlist/Algebraic Groups/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). 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 $x$ of complex manifolds parametrized by a complex space $5$ is defined as any smooth (i.e. locally structured as a projection of a direct product with smooth factors) analytic mapping $\pi : X \rightarrow S$. If $5$ is connected, then all fibres $X _ { S }$, $s \in S$, of $31$ are diffeomorphic to a fixed fibre $X$ where $o \in S$, and may be considered as a family of complex structures on $X$, analytically dependent on the parameter $s \in S$. If the fibres of the family $x$ consist all of complex manifolds diffeomorphic to $X$ and if all fibres are pairwise non-isomorphic, $5$ is said to be the moduli space of the real manifold $X$. 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). Similar, though incomplete, results were also obtained for compact manifolds of complex dimension 2 (cf. 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 , ,  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 $X$ is an analytic family $\pi : X \rightarrow S$, where $5$ is a complex space with a marked point $[ 7 ]$, with the fibre over it coinciding with $X$. The deformation $X = X _ { 0 } \times S$ is said to be trivial. A deformation $\pi : X \rightarrow S$ of the manifold $X$ is said to be isomorphic to the deformation $\pi : X \rightarrow S$ if there exists an analytic isomorphism $\phi : \tilde { X } \rightarrow X$ which is the identity on $X$ and which is such that $\pi \circ \phi = \tilde { \pi }$. If $\pi : X \rightarrow S$ is an analytic deformation, then any analytic mapping $f : S ^ { \prime } \rightarrow S$, where $S ^ { \prime }$ is a space with a marked point $0 ^ { \prime }$ and $f ( \mathfrak { o } ^ { \prime } ) = \mathfrak { o }$, defines by a base change a deformation $X \times S S ^ { \prime } \rightarrow S ^ { \prime }$; the inverse image of this deformation under the mapping $f$. The deformation $\pi : X \rightarrow S$ is said to be locally complete (at the point $[ 7 ]$) if any analytic deformation $\pi ^ { \prime } : X ^ { \prime } \rightarrow S ^ { \prime }$ of the manifold $X$ is isomorphic in some neighbourhood of the marked point to its inverse image for some local analytic mapping $f : S ^ { \prime } \rightarrow S$. If $d f _ { 0 } ^ { \prime }$ is unambiguously defined, the deformation is said to be versal at $[ 7 ]$, and if the germ of the mapping $f$ is uniquely defined, the deformation is said to be universal. An important role in the theory is played by the linear mapping $T _ { \emptyset } ( S ) \rightarrow H ^ { 1 } ( X _ { \diamond } , \Theta )$, where $\Theta = \Theta _ { X _ { 0 } }$ is the sheaf of germs of holomorphic vector fields on $X$, 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 , states that for each compact complex manifold $X$ there exists a deformation versal at the point $[ 7 ]$, which is parametrized by a (not necessarily smooth) analytic subspace $5$ in a neighbourhood of zero of the space $H ^ { 1 } ( X , \Theta )$. Here $5$ is the fibre at the point $[ 7 ]$ of some local analytic mapping $\gamma : H ^ { 1 } ( X _ { 0 } , \Theta ) \rightarrow H ^ { 2 } ( X _ { 0 } , \Theta )$ of the form $\gamma ( \xi ) = [ \xi , \xi ] + \ldots$, where $[ , ]$ is the operation in the graded Lie algebra $H ^ { * } ( X _ { \diamond } , \Theta )$ induced by the Lie bracket in the sheaf $( n$, the dots denoting terms of order 3 or higher. If $H ^ { 1 } ( X _ { 0 } , \Theta ) = 0$, the manifold $X$ is rigid, i.e. any deformation of it is locally trivial (the Fröhlicher–Nijenhuis rigidity theorem). If $H ^ { 2 } ( X , \Theta ) = 0$, $5$ is a neighbourhood of zero in $H ^ { 1 } ( X , \Theta )$. The tangent space $T _ { 0 } ( S )$ always coincides with $H ^ { 1 } ( X , \Theta )$. A deformation is complete at the point $[ 7 ]$ if and only if the corresponding infinitesimal deformation is surjective, and versality is equivalent to bijectivity of the infinitesimal deformation. If $H ^ { 0 } ( X _ { s } , \Theta _ { X _ { S } } )$, $s \in S$, is constant in a neighbourhood of $[ 7 ]$, 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 $\pi : X \rightarrow S$ be smooth and that the fibres be compact are then replaced by the requirements that $31$ be a proper flat mapping. Here, too, it is possible to prove the existence of a deformation versal at the point $[ 7 ]$ , , .

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 .

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 (bundle) $k$ with compact base $x$ and complex Lie group $k$ as structure group there exists a deformation of $k$, versal at a point $[ 7 ]$, parametrized by an analytic subspace in a neighbourhood of zero of the space $H ^ { 1 } ( X , O _ { Ad } _ { E } )$ where $O _ { Ad } _ { E }$ is the sheaf of germs of holomorphic sections of the vector bundle over $x$ associated with $k$ by the adjoint representation . If $x$ is a compact Riemann surface and $k$ 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 $x$ is an arbitrary complex space of finite dimension, a flat analytic family of compact analytic subspaces of $x$ (i.e. an analytic subspace $Y \subseteq X \times S$, where $5$ is a complex space and the projection $Y \rightarrow S$ is a proper flat mapping) has been constructed , and is a universal (in the global sense) deformation for any compact analytic subspace of $x$. In particular, $5$ 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 $\Omega$ 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 $X$ the set $D ( S )$ consists of classes of locally isomorphic deformations of the space $X$ parametrized by an analytic space germ $5$. If $5$ and an element $\delta \in D ( S )$ are fixed, there results a morphism of functors $( . S ) \rightarrow D$. The surjectivity of this morphism (the pair $( S , \delta )$ is said to be complete in such a case) corresponds to the property of completeness of the deformation $0$, 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 $\Omega$. This stimulated the study of covariant functors from the category of Artinian rings into the category of sets satisfying certain natural conditions . 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, 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 .

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Maximilian Janisch/latexlist/Algebraic Groups/Deformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Maximilian_Janisch/latexlist/Algebraic_Groups/Deformation&oldid=43996