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A [[Complex space|complex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h0475701.png" /> that satisfies the following condition: For each compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h0475702.png" /> the set
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<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/h/h047/h047570/h0475703.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h0475704.png" /> is the algebra of holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h0475705.png" />, is compact. A space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h0475706.png" /> is holomorphically convex if and only if it admits a proper surjective holomorphic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h0475707.png" /> onto some [[Stein space|Stein space]] (a holomorphically-complete space) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h0475708.png" /> which induces an isomorphism between the algebras of holomorphic functions on these spaces. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h0475709.png" /> (the holomorphic reduction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757010.png" />) is uniquely defined and has connected fibres [[#References|[1]]]. For any [[Coherent analytic sheaf|coherent analytic sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757011.png" /> on a holomorphically-convex complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757012.png" />, the cohomology spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757015.png" />, are separable topological vector spaces [[#References|[2]]].
+
A [[Complex space|complex space]] $  X $
 +
that satisfies the following condition: For each compactum  $  K \subset  X $
 +
the set
  
A special class of holomorphically-convex complex spaces is formed by the complex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757016.png" /> for which the holomorphic reduction mapping is bijective outside some compact analytic set (such a space is obtained from a Stein space by a proper [[Modification|modification]] which blows-up a finite number of points). A complex space possesses this property if and only if
+
$$
 +
\left \{ {x \in X } : {| f ( x) | \leq
 +
\sup _ { K }  | f | ( f \in A) } \right \}
 +
,
 +
$$
  
<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/h/h047/h047570/h04757017.png" /></td> </tr></table>
+
where  $  A $
 +
is the algebra of holomorphic functions on  $  X $,
 +
is compact. A space  $  X $
 +
is holomorphically convex if and only if it admits a proper surjective holomorphic mapping  $  \phi $
 +
onto some [[Stein space|Stein space]] (a holomorphically-complete space)  $  \widetilde{X}  $
 +
which induces an isomorphism between the algebras of holomorphic functions on these spaces. The mapping  $  \phi : X \rightarrow \widetilde{X}  $(
 +
the holomorphic reduction of  $  X $)
 +
is uniquely defined and has connected fibres [[#References|[1]]]. For any [[Coherent analytic sheaf|coherent analytic sheaf]]  $  F $
 +
on a holomorphically-convex complex space  $  X $,
 +
the cohomology spaces  $  H  ^ {p} ( X, F  ) $
 +
and  $  H _ {c}  ^ {p} ( X, F  ) $,
 +
$  p \geq  0 $,
 +
are separable topological vector spaces [[#References|[2]]].
  
for any coherent analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757019.png" /> [[#References|[3]]]. This class of complex spaces also coincides with the class of strictly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757021.png" />-convex complex spaces (cf. [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]).
+
A special class of holomorphically-convex complex spaces is formed by the complex spaces  $  X $
 +
for which the holomorphic reduction mapping is bijective outside some compact analytic set (such a space is obtained from a Stein space by a proper [[Modification|modification]] which blows-up a finite number of points). A complex space possesses this property if and only if
 +
 
 +
$$
 +
\mathop{\rm dim}  H  ^ {p} ( X, F  )  <  \infty ,\ \
 +
p > 0,
 +
$$
 +
 
 +
for any coherent analytic sheaf $  F $
 +
on $  X $[[#References|[3]]]. This class of complex spaces also coincides with the class of strictly $  1 $-
 +
convex complex spaces (cf. [[Pseudo-convex and pseudo-concave|Pseudo-convex and pseudo-concave]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  "Quotients of complex analytic spaces" , ''Contributions to function theory. Internat. Colloq. Function Theory, Bombay 1960'' , Tata Inst.  (1960)  pp. 1–15</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.P. Ramis,  "Théorèmes de séperation et de finitude pour l'homologie et la cohomologie des espaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757022.png" />-convexes-concaves"  ''Ann. Scuola Norm. Sup. Pisa Ser. 3'' , '''27'''  (1973)  pp. 933–997</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Narasimhan,  "The Levi problem for complex spaces II"  ''Math. Ann.'' , '''146'''  (1962)  pp. 195–216</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  "Quotients of complex analytic spaces" , ''Contributions to function theory. Internat. Colloq. Function Theory, Bombay 1960'' , Tata Inst.  (1960)  pp. 1–15</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.P. Ramis,  "Théorèmes de séperation et de finitude pour l'homologie et la cohomologie des espaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047570/h04757022.png" />-convexes-concaves"  ''Ann. Scuola Norm. Sup. Pisa Ser. 3'' , '''27'''  (1973)  pp. 933–997</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  R. Narasimhan,  "The Levi problem for complex spaces II"  ''Math. Ann.'' , '''146'''  (1962)  pp. 195–216</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)  pp. Chapt. 1, Sect. G</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Komplexe Räume"  ''Math. Ann.'' , '''136'''  (1958)  pp. 245–318</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gunning,  H. Rossi,  "Analytic functions of several complex variables" , Prentice-Hall  (1965)  pp. Chapt. 1, Sect. G</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Theory of Stein spaces" , Springer  (1979)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Grauert,  R. Remmert,  "Komplexe Räume"  ''Math. Ann.'' , '''136'''  (1958)  pp. 245–318</TD></TR></table>

Latest revision as of 22:10, 5 June 2020


A complex space $ X $ that satisfies the following condition: For each compactum $ K \subset X $ the set

$$ \left \{ {x \in X } : {| f ( x) | \leq \sup _ { K } | f | ( f \in A) } \right \} , $$

where $ A $ is the algebra of holomorphic functions on $ X $, is compact. A space $ X $ is holomorphically convex if and only if it admits a proper surjective holomorphic mapping $ \phi $ onto some Stein space (a holomorphically-complete space) $ \widetilde{X} $ which induces an isomorphism between the algebras of holomorphic functions on these spaces. The mapping $ \phi : X \rightarrow \widetilde{X} $( the holomorphic reduction of $ X $) is uniquely defined and has connected fibres [1]. For any coherent analytic sheaf $ F $ on a holomorphically-convex complex space $ X $, the cohomology spaces $ H ^ {p} ( X, F ) $ and $ H _ {c} ^ {p} ( X, F ) $, $ p \geq 0 $, are separable topological vector spaces [2].

A special class of holomorphically-convex complex spaces is formed by the complex spaces $ X $ for which the holomorphic reduction mapping is bijective outside some compact analytic set (such a space is obtained from a Stein space by a proper modification which blows-up a finite number of points). A complex space possesses this property if and only if

$$ \mathop{\rm dim} H ^ {p} ( X, F ) < \infty ,\ \ p > 0, $$

for any coherent analytic sheaf $ F $ on $ X $[3]. This class of complex spaces also coincides with the class of strictly $ 1 $- convex complex spaces (cf. Pseudo-convex and pseudo-concave).

References

[1] H. Cartan, "Quotients of complex analytic spaces" , Contributions to function theory. Internat. Colloq. Function Theory, Bombay 1960 , Tata Inst. (1960) pp. 1–15
[2] J.P. Ramis, "Théorèmes de séperation et de finitude pour l'homologie et la cohomologie des espaces -convexes-concaves" Ann. Scuola Norm. Sup. Pisa Ser. 3 , 27 (1973) pp. 933–997
[3] R. Narasimhan, "The Levi problem for complex spaces II" Math. Ann. , 146 (1962) pp. 195–216

Comments

References

[a1] R.C. Gunning, H. Rossi, "Analytic functions of several complex variables" , Prentice-Hall (1965) pp. Chapt. 1, Sect. G
[a2] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1979) (Translated from German)
[a3] H. Grauert, R. Remmert, "Komplexe Räume" Math. Ann. , 136 (1958) pp. 245–318
How to Cite This Entry:
Holomorphically-convex complex space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Holomorphically-convex_complex_space&oldid=16349
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article