Real-analytic space

An analytic space over the field $\mathbf R$ of real numbers. Unlike in the case of complex-analytic spaces, the structure sheaves of real-analytic spaces need not be coherent (cf. Coherent sheaf). Real-analytic spaces are said to be coherent if their structure sheaves are coherent. All real-analytic manifolds (i.e. smooth real-analytic spaces) are coherent real-analytic spaces.

Let $V _ {a}$ be the germ at a point $a$ of a real-analytic subset of $\mathbf R ^ {n}$ (cf. Analytic set). This defines the germ at $a$ of a complex-analytic subset $\widetilde{V} _ {a}$ of the space $\mathbf C ^ {n}$ with the following equivalent properties: 1) $\widetilde{V}_ {a}$ is the intersection of all germs of complex-analytic sets containing $V _ {a}$; 2) if ${\mathcal O} _ {V _ {a} }$ is the analytic algebra of the germ $V _ {a}$, then ${\mathcal O} _ {V _ {a} } \otimes \mathbf C$ is the analytic algebra of the germ $\widetilde{V}_ {a}$. The germ $\widetilde{V}_ {a}$ is said to be the complexification of the germ $V _ {a}$, and $V _ {a}$ is said to be the real part of the germ $\widetilde{V}_ {a}$. Analogously, for any coherent real-analytic countably-infinite space $X$ it is possible to construct the complexification $\widetilde{X}$, which is a complex-analytic space. $X$ will then have a fundamental system of neighbourhoods in $\widetilde{X}$ which are Stein spaces (cf. Stein space).

The theory of coherent real-analytic spaces is similar to the theory of complex Stein spaces. Global sections of any coherent analytic sheaf of modules $F$ on a coherent real-analytic countably-infinite space $X$ generate modules of germs of its sections at any point of $X$, and all groups $H ^ {q} ( X, F )$ vanish if $q \geq 1$.

For any finite-dimensional coherent real-analytic countably-infinite space $( X, {\mathcal O} _ {X} )$ there exists a morphism

$$f = ( f _ {0} , f _ {1} ): ( X, {\mathcal O} _ {X} ) \rightarrow \ ( \mathbf R ^ {n} , {\mathcal O} _ {\mathbf R ^ {n} } )$$

such that $f _ {0}$ is a proper one-to-one mapping of $X$ into a coherent subspace in $\mathbf R ^ {n}$, while $f$ is an imbedding at the smooth points of $X$. In particular, any (Hausdorff and countably-infinite) real-analytic manifold is isomorphic to a real-analytic submanifold in $\mathbf R ^ {n}$. For a reduced coherent real-analytic space $X$ the set of classes of isomorphic real-analytic principal fibre bundles with a real structure Lie group $G$, admitting complexification, and base $X$ is in one-to-one correspondence with the set of classes of isomorphic topological principal fibre bundles with the same structure group $G$.

References

 [1] A. Tognoli, "Some results in the theory of real analytic spaces" M. Jurchesan (ed.) , Espaces Analytiques (Bucharest 1969) , Acad. Roumanie (1971) pp. 149–157