Analytic manifold

A manifold with an analytic atlas. The structure of an $n$-dimensional manifold $M$ over a complete non-discretely normed field $k$ on a topological space is defined by specifying an analytic atlas over $k$ on $M$, i.e. a collection of charts (cf. Chart) with values in $k^n$ covering $M$, any two charts of which are analytically related. Two atlases are said to define the same structure if their union is an analytic atlas. The sheaf $\mathcal O$ of germs of $k$-valued analytic functions is defined on an analytic manifold. The class of ringed spaces $(M,\mathcal O)$ which results in this way is identical with the class of smooth analytic spaces over $k$.

If $k$ is the field of real numbers $\mathbf R$ one speaks of real-analytic manifolds; if $k$ is the field of complex numbers $\mathbf C$, of complex-analytic or simply complex manifolds; if $k$ is the field of $p$-adic numbers $\mathbf Q_p$, of $p$-adic analytic manifolds. Examples of analytic manifolds include the $n$-dimensional Euclidean spaces $k^n$, the $n$-dimensional projective spaces over $k$, the affine and projective algebraic varieties without singular points over $k$, and Lie groups and their homogeneous spaces.

The concept of an analytic manifold goes back to B. Riemann and F. Klein, but was precisely formulated for the first time by H. Weyl [4] for the case of Riemann surfaces, i.e. one-dimensional complex manifolds. At present (the 1970's) it is natural to regard analytic manifolds as a special case of analytic spaces (cf. Analytic space), which may be roughly described as "varieties with singular points" . The concept of an analytic space was introduced in the 1950's and became the principal subject of the theory of analytic functions; many fundamental results obtained for analytic manifolds could be successfully applied to the non-smooth case. For an account of the general properties of analytic manifolds over an arbitrary field see [3].

There is a close relationship between the theories of real-analytic and differentiable manifolds (cf. Differentiable manifold), and also between the theories of real-analytic and complex-analytic manifolds, Clearly, the natural structure of a manifold of class $C^\infty$ is defined on each real-analytic manifold. It was shown by H. Whitney in 1936 that the converse proposition is also true: It is possible to define on any paracompact manifold of class $C^\infty$ an analytic structure over $\mathbf R$ which induces the initial smooth structure. It follows from Grauert's theorem on the imbeddability of a paracompact analytic manifold over $\mathbf R$ in a Euclidean space that this analytic structure is unambiguously defined up to an isomorphism (not necessarily the identity) [2].

A natural structure of a real-analytic manifold (of double dimension) is defined on all complex manifolds $M$. The answer to the converse problem — viz. whether a complex structure on a given real-analytic manifold exists and whether it is unique — has been given in the simplest cases only. Thus, if $M$ is a connected two-dimensional real-analytic manifold, then a necessary and sufficient condition for the existence of a complex structure on $M$ is paracompactness and orientability, while the problem of classification of these structures is identical with the classical moduli problem for Riemann surfaces (cf. Moduli of a Riemann surface). There is a classification of compact analytic surfaces (i.e. of two-dimensional complex manifolds, cf. Analytic surface), which gives a partial answer to the above problem for four-dimensional real-analytic manifolds. On the other hand it is possible, using topological methods, to identify classes of real manifolds that do not permit almost-complex or, a fortiori, complex structures. Such manifolds include the spheres $S^{2k}$ for $k\neq1,3$. A description of complex structures which are sufficiently near to a given complex structure is given by the theory of deformations of analytic structures (cf. Deformation), in which an important role is played by Banach analytic manifolds — infinite-dimensional analogues of analytic manifolds.

References

[1]	N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French) MR0205211 MR0205210
[2]	R. Narasimhan, "Analysis on real and complex manifolds" , Springer (1971) MR0832683 MR0346855 MR0251745 Zbl 0583.58001 Zbl 0188.25803
[3]	J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) MR0218496 Zbl 0132.27803
[4]	H. Weyl, "Die Idee der Riemannschen Fläche" , Teubner (1955) MR0069903 Zbl 0068.06001

Comments

A much related basic problem in complex analysis is the question whether there are any complex structures on projective space besides the usual one (and inducing the same topology). For $n=1$ this is very classical (all Riemann surfaces of genus zero are isomorphic to $P_\mathbf C^1$). For $n=2$, uniqueness of the complex structure follows from combined work of F. Hirzebruch, K. Kodaira [a4], and S.T. Yau [a5]. For $n=3$ one has that a compact manifold that is bimeromorphically equivalent to a Kähler manifold and that is also topologically $P_\mathbf C^3$ is analytically isomorphic to $P_\mathbf C^3$ [a6].

References

[a1]	H. Whitney, "Complex analytic varieties" , Addison-Wesley (1972) MR0387634 Zbl 0265.32008
[a2]	W. Barth, C. Peters, A. van der Ven, "Compact complex surfaces" , Springer (1984) MR0749574 Zbl 0718.14023
[a3]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004
[a4]	F. Hirzebruch, K. Kodaira, "On the complex projective spaces" J. Math. Pures Appl. , 36 (1957) pp. 201–216 MR0092195 Zbl 0090.38601
[a5]	S.-T. Yau, "Calabi's conjecture and some new results in algebraic geometry" Proc. Nat. Acad. Sci. USA , 74 (1977) pp. 1798–1799
[a6]	T. Peternell, "A rigidity theorem for $P_3(\mathbf C)$" Manuscripta Math. , 50 (1985) pp. 397–428 MR0784150 Zbl 0573.32027