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A generalization of the concept of an ordinary surface in three-dimensional space to the case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485201.png" />-dimensional space. The dimension of a hypersurface is one less than that of its ambient space.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485203.png" /> are differentiable manifolds, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485204.png" />, and if an immersion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485205.png" /> has been defined, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485206.png" /> is a hypersurface in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485207.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485208.png" /> is a differentiable mapping whose differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h0485209.png" /> at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852010.png" /> is an injective mapping of the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852011.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852012.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852013.png" /> into the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852014.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852016.png" />.
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 +
A generalization of the concept of an ordinary surface in three-dimensional space to the case of an  $  n $-
 +
dimensional space. The dimension of a hypersurface is one less than that of its ambient space.
  
 +
If  $  M $
 +
and  $  N $
 +
are differentiable manifolds,  $  \mathop{\rm dim}  N -  \mathop{\rm dim}  M = 1 $,
 +
and if an immersion  $  f:  M \rightarrow N $
 +
has been defined, then  $  f( M) $
 +
is a hypersurface in  $  N $.
 +
Here  $  f $
 +
is a differentiable mapping whose differential  $  df $
 +
at any point  $  x \in M $
 +
is an injective mapping of the tangent space  $  M _ {x} $
 +
to  $  M $
 +
at  $  x $
 +
into the tangent space  $  N _ {f(} x) $
 +
to  $  N $
 +
at  $  f( x) $.
  
 
====Comments====
 
====Comments====
  
 +
====References====
 +
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Spivak, "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish (1975) pp. 1–5 {{MR|0394453}} {{MR|0394452}} {{MR|0372756}} {{ZBL|0306.53003}} {{ZBL|0306.53002}} {{ZBL|0306.53001}} </TD></TR></table>
  
====References====
+
An algebraic hypersurface is a subvariety of an algebraic variety that is locally defined by one equation. An algebraic hypersurface in the affine space  $ A _ {k}  ^ {n} $
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Spivak,  "A comprehensive introduction to differential geometry" , '''1979''' , Publish or Perish (1975) pp. 1–5</TD></TR></table>
+
over a field $ k $
 +
is globally defined by one equation
  
An algebraic hypersurface is a subvariety of an algebraic variety that is locally defined by one equation. An algebraic hypersurface in the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852017.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852018.png" /> is globally defined by one equation
+
$$
 +
f ( x _ {1} \dots x _ {n} )  = 0.
 +
$$
  
<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/h048/h048520/h04852019.png" /></td> </tr></table>
+
An algebraic hypersurface  $  W $
 +
in a projective space  $  P  ^ {n} $
 +
is defined by an equation
  
An algebraic hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852020.png" /> in a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852021.png" /> is defined by an equation
+
$$
 +
F ( x _ {0} \dots x _ {n} )  = 0,
 +
$$
  
<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/h048/h048520/h04852022.png" /></td> </tr></table>
+
where  $  F $
 +
is a homogeneous form in  $  n + 1 $
 +
variables. The degree  $  m $
 +
of this form is said to be the degree (order) of the hypersurface. A closed subscheme  $  W $
 +
of a scheme  $  V $
 +
is said to be a hypersurface if the corresponding sheaf of ideals  $  I _ {W} \subset  {\mathcal O} _ {V} $
 +
is a sheaf of principal ideals. For a connected non-singular algebraic variety this condition means that the codimension of  $  W $
 +
in  $  V $
 +
is one. For each non-singular algebraic hypersurface  $  W \subset  P _ {k}  ^ {n} $
 +
of order  $  m $(
 +
often denoted by  $  V _ {n}  ^ {m} $)
 +
the following holds:
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852023.png" /> is a homogeneous form in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852024.png" /> variables. The degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852025.png" /> of this form is said to be the degree (order) of the hypersurface. A closed subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852026.png" /> of a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852027.png" /> is said to be a hypersurface if the corresponding sheaf of ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852028.png" /> is a sheaf of principal ideals. For a connected non-singular algebraic variety this condition means that the codimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852030.png" /> is one. For each non-singular algebraic hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852031.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852032.png" /> (often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852033.png" />) the following holds:
+
a) the canonical class  $  K _ {W} $
 +
is equal to ( m - n - 1 ) H _ {W} $
 +
where  $  H _ {W} $
 +
is the class of a hyperplane section of $  W $;
  
a) the canonical class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852034.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852035.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852036.png" /> is the class of a hyperplane section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852037.png" />;
+
b) the cohomology groups  $  H  ^ {i} ( W, {\mathcal O} _ {W} ) = 0 $
 +
for  $  i \neq 0, n - 1 $,
 +
and
  
b) the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852038.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852039.png" />, and
+
$$
 +
\mathop{\rm dim} _ {k}  H ^ {n - 1 } ( W, {\mathcal O} _ {W} ) = \
  
<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/h048/h048520/h04852040.png" /></td> </tr></table>
+
\frac{( m - 1) \dots ( m - n) }{n! }
 +
;
 +
$$
  
c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852041.png" />, the fundamental group (algebraic or topological if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852042.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852043.png" />;
+
c) if $  n \geq  3 $,  
 +
the fundamental group (algebraic or topological if $  k = \mathbf C $)
 +
$  \pi _ {1} ( W) = 0 $;
  
d) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852044.png" />, the Picard group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852045.png" /> and is generated by the class of a hyperplane section.
+
d) if $  n \geq  4 $,  
 +
the Picard group $  \mathop{\rm Pic} ( W) \simeq \mathbf Z $
 +
and is generated by the class of a hyperplane section.
  
 
''I.V. Dolgachev''
 
''I.V. Dolgachev''
Line 37: Line 94:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Carlson,   P. Griffiths,   "Infinitesimal variations of Hodge structure and the global Torelli problem" A. Beauville (ed.) , ''Algebraic geometry (Angers, 1979)'' , Sijthoff &amp; Noordhoff (1980) pp. 51–76</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Donagi,   "Generic Torelli for projective hypersurfaces" ''Compos. Math.'' , '''50''' (1983) pp. 325–353</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Carlson, P. Griffiths, "Infinitesimal variations of Hodge structure and the global Torelli problem" A. Beauville (ed.) , ''Algebraic geometry (Angers, 1979)'' , Sijthoff &amp; Noordhoff (1980) pp. 51–76 {{MR|0605336}} {{ZBL|0479.14007}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Donagi, "Generic Torelli for projective hypersurfaces" ''Compos. Math.'' , '''50''' (1983) pp. 325–353 {{MR|0720291}} {{ZBL|0598.14007}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
  
An analytic hypersurface is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852046.png" /> in a complex Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852047.png" /> that, in a neighbourhood of each of its points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852048.png" />, is defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852049.png" />, where the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852050.png" /> is continuous with respect to the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852052.png" />, and, for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852053.png" />, is holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852054.png" /> in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852055.png" /> which is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852056.png" />; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852057.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852058.png" />. In other words, an analytic hypersurface is a set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852059.png" /> that is locally the union of a continuous one-parameter family of complex-analytic surfaces of complex codimension one. For instance, if a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852060.png" /> is holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852063.png" />, then the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852065.png" />, etc., are analytic hypersurfaces.
+
An analytic hypersurface is a set $  S $
 +
in a complex Euclidean space $  \mathbf C  ^ {n} $
 +
that, in a neighbourhood of each of its points $  \zeta \in S $,  
 +
is defined by an equation $  f _  \zeta  ( z, t) = 0 $,  
 +
where the function $  f _  \zeta  ( z, t) $
 +
is continuous with respect to the parameter $  t \in ( - \epsilon , \epsilon ) $,
 +
$  \epsilon > 0 $,  
 +
and, for each fixed $  t $,  
 +
is holomorphic in $  z $
 +
in a neighbourhood $  U _  \zeta  \ni \zeta $
 +
which is independent of $  t $;  
 +
moreover, $  \sum | \partial  f/ \partial  z _ {j} | \neq 0 $
 +
for all $  ( z, t) \in U _  \zeta  \times ( - \epsilon , \epsilon ) $.  
 +
In other words, an analytic hypersurface is a set in $  \mathbf C  ^ {n} $
 +
that is locally the union of a continuous one-parameter family of complex-analytic surfaces of complex codimension one. For instance, if a function $  f $
 +
is holomorphic in a domain $  D \subset  \mathbf C  ^ {n} $
 +
and $  \mathop{\rm grad}  f \neq 0 $
 +
in $  D $,  
 +
then the sets $  | f | = 1 $,  
 +
$  \mathop{\rm Re}  f = 0 $,  
 +
etc., are analytic hypersurfaces.
  
A twice-differentiable hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852066.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852067.png" /> is an analytic hypersurface if and only if its Levi form vanishes identically on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852068.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852069.png" /> is locally pseudo-convex on both sides.
+
A twice-differentiable hypersurface $  S $
 +
in $  \mathbf R  ^ {2n} = \mathbf C  ^ {n} $
 +
is an analytic hypersurface if and only if its Levi form vanishes identically on $  S $
 +
or if $  S $
 +
is locally pseudo-convex on both sides.
  
 
''E.M. Chirka''
 
''E.M. Chirka''
  
 
====Comments====
 
====Comments====
Sometimes the phrase "analytic hypersurface" is also used for an [[Analytic set|analytic set]] of complex codimension 1, analogously to 3) above, cf. [[#References|[a1]]]. An analytic hypersurface as in 4) is also called a foliation by analytic varieties of codimension 1. The result concerning a twice-differentiable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048520/h04852070.png" />, mentioned above, can be found in [[#References|[a2]]].
+
Sometimes the phrase "analytic hypersurface" is also used for an [[Analytic set|analytic set]] of complex codimension 1, analogously to 3) above, cf. [[#References|[a1]]]. An analytic hypersurface as in 4) is also called a foliation by analytic varieties of codimension 1. The result concerning a twice-differentiable $  S \subset  \mathbf R  ^ {2n} $,  
 +
mentioned above, can be found in [[#References|[a2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert,   R. Remmert,   "Theory of Stein spaces" , Springer (1977) (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Vladimirov,   "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Kaup,   B. Kaup,   "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1977) (Translated from German) {{MR|0513229}} {{ZBL|0379.32001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L. Kaup, B. Kaup, "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German) {{MR|0716497}} {{ZBL|0528.32001}} </TD></TR></table>

Latest revision as of 22:11, 5 June 2020


A generalization of the concept of an ordinary surface in three-dimensional space to the case of an $ n $- dimensional space. The dimension of a hypersurface is one less than that of its ambient space.

If $ M $ and $ N $ are differentiable manifolds, $ \mathop{\rm dim} N - \mathop{\rm dim} M = 1 $, and if an immersion $ f: M \rightarrow N $ has been defined, then $ f( M) $ is a hypersurface in $ N $. Here $ f $ is a differentiable mapping whose differential $ df $ at any point $ x \in M $ is an injective mapping of the tangent space $ M _ {x} $ to $ M $ at $ x $ into the tangent space $ N _ {f(} x) $ to $ N $ at $ f( x) $.

Comments

References

[a1] M. Spivak, "A comprehensive introduction to differential geometry" , 1979 , Publish or Perish (1975) pp. 1–5 MR0394453 MR0394452 MR0372756 Zbl 0306.53003 Zbl 0306.53002 Zbl 0306.53001

An algebraic hypersurface is a subvariety of an algebraic variety that is locally defined by one equation. An algebraic hypersurface in the affine space $ A _ {k} ^ {n} $ over a field $ k $ is globally defined by one equation

$$ f ( x _ {1} \dots x _ {n} ) = 0. $$

An algebraic hypersurface $ W $ in a projective space $ P ^ {n} $ is defined by an equation

$$ F ( x _ {0} \dots x _ {n} ) = 0, $$

where $ F $ is a homogeneous form in $ n + 1 $ variables. The degree $ m $ of this form is said to be the degree (order) of the hypersurface. A closed subscheme $ W $ of a scheme $ V $ is said to be a hypersurface if the corresponding sheaf of ideals $ I _ {W} \subset {\mathcal O} _ {V} $ is a sheaf of principal ideals. For a connected non-singular algebraic variety this condition means that the codimension of $ W $ in $ V $ is one. For each non-singular algebraic hypersurface $ W \subset P _ {k} ^ {n} $ of order $ m $( often denoted by $ V _ {n} ^ {m} $) the following holds:

a) the canonical class $ K _ {W} $ is equal to $ ( m - n - 1 ) H _ {W} $ where $ H _ {W} $ is the class of a hyperplane section of $ W $;

b) the cohomology groups $ H ^ {i} ( W, {\mathcal O} _ {W} ) = 0 $ for $ i \neq 0, n - 1 $, and

$$ \mathop{\rm dim} _ {k} H ^ {n - 1 } ( W, {\mathcal O} _ {W} ) = \ \frac{( m - 1) \dots ( m - n) }{n! } ; $$

c) if $ n \geq 3 $, the fundamental group (algebraic or topological if $ k = \mathbf C $) $ \pi _ {1} ( W) = 0 $;

d) if $ n \geq 4 $, the Picard group $ \mathop{\rm Pic} ( W) \simeq \mathbf Z $ and is generated by the class of a hyperplane section.

I.V. Dolgachev

Comments

The cohomology ring of a smooth complex projective hypersurface can be expressed completely in terms of rational differential forms on the ambient projective space, [a1]. In most cases, the period mapping for these hypersurfaces has been shown to be of degree one [a2].

References

[a1] J. Carlson, P. Griffiths, "Infinitesimal variations of Hodge structure and the global Torelli problem" A. Beauville (ed.) , Algebraic geometry (Angers, 1979) , Sijthoff & Noordhoff (1980) pp. 51–76 MR0605336 Zbl 0479.14007
[a2] R. Donagi, "Generic Torelli for projective hypersurfaces" Compos. Math. , 50 (1983) pp. 325–353 MR0720291 Zbl 0598.14007
[a3] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 13; 170; 316; 381 MR0463157 Zbl 0367.14001

An analytic hypersurface is a set $ S $ in a complex Euclidean space $ \mathbf C ^ {n} $ that, in a neighbourhood of each of its points $ \zeta \in S $, is defined by an equation $ f _ \zeta ( z, t) = 0 $, where the function $ f _ \zeta ( z, t) $ is continuous with respect to the parameter $ t \in ( - \epsilon , \epsilon ) $, $ \epsilon > 0 $, and, for each fixed $ t $, is holomorphic in $ z $ in a neighbourhood $ U _ \zeta \ni \zeta $ which is independent of $ t $; moreover, $ \sum | \partial f/ \partial z _ {j} | \neq 0 $ for all $ ( z, t) \in U _ \zeta \times ( - \epsilon , \epsilon ) $. In other words, an analytic hypersurface is a set in $ \mathbf C ^ {n} $ that is locally the union of a continuous one-parameter family of complex-analytic surfaces of complex codimension one. For instance, if a function $ f $ is holomorphic in a domain $ D \subset \mathbf C ^ {n} $ and $ \mathop{\rm grad} f \neq 0 $ in $ D $, then the sets $ | f | = 1 $, $ \mathop{\rm Re} f = 0 $, etc., are analytic hypersurfaces.

A twice-differentiable hypersurface $ S $ in $ \mathbf R ^ {2n} = \mathbf C ^ {n} $ is an analytic hypersurface if and only if its Levi form vanishes identically on $ S $ or if $ S $ is locally pseudo-convex on both sides.

E.M. Chirka

Comments

Sometimes the phrase "analytic hypersurface" is also used for an analytic set of complex codimension 1, analogously to 3) above, cf. [a1]. An analytic hypersurface as in 4) is also called a foliation by analytic varieties of codimension 1. The result concerning a twice-differentiable $ S \subset \mathbf R ^ {2n} $, mentioned above, can be found in [a2].

References

[a1] H. Grauert, R. Remmert, "Theory of Stein spaces" , Springer (1977) (Translated from German) MR0513229 Zbl 0379.32001
[a2] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)
[a3] L. Kaup, B. Kaup, "Holomorphic functions of several variables" , de Gruyter (1983) (Translated from German) MR0716497 Zbl 0528.32001
How to Cite This Entry:
Hypersurface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypersurface&oldid=14567
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article