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A generalization of the concept of a [[Divisor|divisor]] of positive degree on a [[Riemann surface|Riemann surface]]. A holomorphic [[Vector bundle|vector bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739801.png" /> over a [[Complex space|complex space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739802.png" /> is called positive (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739803.png" />) if on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739804.png" /> there exists a [[Hermitian metric|Hermitian metric]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739805.png" /> such that the function
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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/p/p073/p073980/p0739806.png" /></td> </tr></table>
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on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739807.png" /> is strictly pseudo-convex outside the zero section. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739808.png" /> is a manifold, the condition for positivity is expressed in terms of the curvature of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p0739809.png" />. That is, the [[Curvature form|curvature form]] of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398010.png" /> in the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398011.png" /> corresponds to a Hermitian quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398012.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398013.png" /> with values in the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398014.png" /> of Hermitian endomorphisms of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398015.png" />. The positivity condition is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398016.png" /> being a positive-definite operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398017.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398018.png" /> and any non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398019.png" />.
+
A generalization of the concept of a [[Divisor|divisor]] of positive degree on a [[Riemann surface|Riemann surface]]. A holomorphic [[Vector bundle|vector bundle]]  $  E $
 +
over a [[Complex space|complex space]]  $  X $
 +
is called positive (denoted by  $  E > 0 $)
 +
if on  $  E $
 +
there exists a [[Hermitian metric|Hermitian metric]]  $  h $
 +
such that the function
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398020.png" /> is a complex line bundle over a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398021.png" />, the condition for positivity is equivalent to that of positive definiteness of the matrix
+
$$
 +
v  \mapsto  - h( v, v)
 +
$$
  
<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/p/p073/p073980/p07398022.png" /></td> </tr></table>
+
on  $  E $
 +
is strictly pseudo-convex outside the zero section. If  $  X $
 +
is a manifold, the condition for positivity is expressed in terms of the curvature of the metric  $  h $.  
 +
That is, the [[Curvature form|curvature form]] of the metric  $  h $
 +
in the bundle  $  E $
 +
corresponds to a Hermitian quadratic form  $  \Omega $
 +
on  $  X $
 +
with values in the bundle  $  \mathop{\rm Herm} ( E) $
 +
of Hermitian endomorphisms of the bundle  $  E $.
 +
The positivity condition is equivalent to  $  \Omega _ {x} ( u) $
 +
being a positive-definite operator on  $  E _ {x} $
 +
for any  $  x \in X $
 +
and any non-zero  $  u \in T _ {X,x }  $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398023.png" /> are local coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398025.png" /> is a function that defines the Hermitian metric for the local trivialization of the bundle. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398026.png" /> is compact, then the complex line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398027.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398028.png" /> is positive if and only if the [[Chern class|Chern class]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398029.png" /> contains a closed form of the type
+
If  $  E $
 +
is a complex line bundle over a manifold  $  X $,
 +
the condition for positivity is equivalent to that of positive definiteness of the matrix
  
<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/p/p073/p073980/p07398030.png" /></td> </tr></table>
+
$$
 +
\left \| -  
 +
\frac{\partial  ^ {2}  \mathop{\rm log}  h }{\partial  z _  \alpha  \partial  {\overline{z}\; } _  \beta  }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398031.png" /> is a positive-definite Hermitian matrix. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398032.png" /> is a Riemann surface, then the bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398033.png" /> defined by a divisor of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398034.png" /> is positive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398035.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398036.png" /> is a bundle of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398037.png" /> over a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398038.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398039.png" />, one can consider also the following narrower class of positive bundles: A bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398040.png" /> is called positive in the sense of Nakano if there exists on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398041.png" /> a Hermitian metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398042.png" /> such that the Hermitian quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398043.png" /> on the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398044.png" /> as defined by the formula
+
\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/p/p073/p073980/p07398045.png" /></td> </tr></table>
+
where  $  z _ {1} \dots z _ {n} $
 +
are local coordinates in  $  X $
 +
and  $  h > 0 $
 +
is a function that defines the Hermitian metric for the local trivialization of the bundle. If  $  X $
 +
is compact, then the complex line bundle  $  E $
 +
over  $  X $
 +
is positive if and only if the [[Chern class|Chern class]]  $  c _ {1} ( E) $
 +
contains a closed form of the type
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398048.png" />, is positive definite. Examples: the tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398049.png" /> of the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398050.png" /> is positive, but for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398051.png" /> it is not positive in the sense of Nakano; the complex line bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398052.png" /> defined by a hyperplane is positive.
+
$$
 +
i \sum _ {\alpha , \beta } \phi _ {\alpha \beta }  dz _  \alpha  d {\overline{z}\; } _  \beta  ,
 +
$$
  
Any quotient bundle of a positive vector bundle is positive. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398054.png" /> are positive bundles (in the sense of Nakano), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398056.png" /> are positive (in the sense of Nakano).
+
where  $  \| \phi _ {\alpha \beta }  \| $
 +
is a positive-definite Hermitian matrix. In particular, if  $  X $
 +
is a Riemann surface, then the bundle over  $  X $
 +
defined by a divisor of degree  $  d $
 +
is positive if and only if  $  d > 0 $.
 +
If  $  E $
 +
is a bundle of rank  $  > 1 $
 +
over a manifold  $  X $
 +
of dimension  $  > 1 $,
 +
one can consider also the following narrower class of positive bundles: A bundle  $  E \rightarrow X $
 +
is called positive in the sense of Nakano if there exists on  $  E $
 +
a Hermitian metric  $  h $
 +
such that the Hermitian quadratic form  $  H $
 +
on the bundle  $  E \otimes T _ {X} $
 +
as defined by the formula
 +
 
 +
$$
 +
H _ {x} ( v \otimes u)  =  h _ {x} ( v, \Omega _ {x} ( u) v),
 +
$$
 +
 
 +
where  $  x \in X $,
 +
$  v \in E _ {x} $
 +
and  $  u \in T _ {X,x }  $,
 +
is positive definite. Examples: the tangent bundle  $  TP _ {n} $
 +
of the projective space  $  P  ^ {n} $
 +
is positive, but for  $  n > 1 $
 +
it is not positive in the sense of Nakano; the complex line bundle over  $  P  ^ {n} $
 +
defined by a hyperplane is positive.
 +
 
 +
Any quotient bundle of a positive vector bundle is positive. If $  E  ^  \prime  $
 +
and $  E  ^ {\prime\prime} $
 +
are positive bundles (in the sense of Nakano), then $  E  ^  \prime  \oplus E  ^ {\prime\prime} $
 +
and $  E  ^  \prime  \otimes E  ^ {\prime\prime} $
 +
are positive (in the sense of Nakano).
  
 
The concept of a positive bundle was introduced in connection with Kodaira's vanishing theorem (cf. [[Kodaira theorem|Kodaira theorem]]) for complex line bundles, and it was then extended to any bundle. Somewhat later, in relation to the existence of an imbedding in a projective space, the concepts of weakly-positive and weakly-negative bundles were introduced.
 
The concept of a positive bundle was introduced in connection with Kodaira's vanishing theorem (cf. [[Kodaira theorem|Kodaira theorem]]) for complex line bundles, and it was then extended to any bundle. Somewhat later, in relation to the existence of an imbedding in a projective space, the concepts of weakly-positive and weakly-negative bundles were introduced.
  
A holomorphic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398057.png" /> over a compact complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398058.png" /> is called weakly negative if its zero section has a strictly pseudo-convex neighbourhood in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398059.png" />, i.e. if it is an [[Exceptional analytic set|exceptional analytic set]]. A bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398060.png" /> is called weakly positive if the dual bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398061.png" /> is weakly negative. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398062.png" /> is a Riemann surface, the concepts of a weakly-positive bundle and a positive bundle coincide [[#References|[5]]]. In the general case, positivity implies weak positivity; no examples are known at present (1983) of weakly positive but non-positive bundles.
+
A holomorphic vector bundle $  E $
 +
over a compact complex space $  X $
 +
is called weakly negative if its zero section has a strictly pseudo-convex neighbourhood in $  E $,  
 +
i.e. if it is an [[Exceptional analytic set|exceptional analytic set]]. A bundle $  E $
 +
is called weakly positive if the dual bundle $  E  ^ {*} $
 +
is weakly negative. If $  X $
 +
is a Riemann surface, the concepts of a weakly-positive bundle and a positive bundle coincide [[#References|[5]]]. In the general case, positivity implies weak positivity; no examples are known at present (1983) of weakly positive but non-positive bundles.
  
Weak positivity of a bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398063.png" /> is equivalent to each of the following properties: For any [[Coherent analytic sheaf|coherent analytic sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398064.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398065.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398066.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398067.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398068.png" /> is generated by global sections; for any analytic sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398069.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398070.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398071.png" /> such that
+
Weak positivity of a bundle $  E \rightarrow X $
 +
is equivalent to each of the following properties: For any [[Coherent analytic sheaf|coherent analytic sheaf]] $  {\mathcal F} $
 +
on $  X $
 +
there exists an $  m _ {0} > 0 $
 +
such that $  {\mathcal F} \otimes S  ^ {m} {\mathcal E} $
 +
for $  m \geq  m _ {0} $
 +
is generated by global sections; for any analytic sheaf $  {\mathcal F} $
 +
on $  X $
 +
there exists an $  m \geq  0 $
 +
such that
  
<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/p/p073/p073980/p07398072.png" /></td> </tr></table>
+
$$
 +
H  ^ {q} ( X, {\mathcal F} \otimes S  ^ {m} {\mathcal E})  = 0
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398073.png" /> [[#References|[3]]], [[#References|[4]]]. By <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398074.png" /> is meant the sheaf of germs of holomorphic sections of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398075.png" />. Weakly-positive bundles are therefore analogous to the ample sheafs (cf. [[Ample sheaf|Ample sheaf]]) from algebraic geometry and are sometimes called ample analytic bundles. A weakly-positive bundle over a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398076.png" /> naturally defines an imbedding of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398077.png" /> into a Grassmann manifold and thus into a projective space.
+
for all $  q \geq  1 $[[#References|[3]]], [[#References|[4]]]. By $  {\mathcal E} $
 +
is meant the sheaf of germs of holomorphic sections of the bundle $  E $.  
 +
Weakly-positive bundles are therefore analogous to the ample sheafs (cf. [[Ample sheaf|Ample sheaf]]) from algebraic geometry and are sometimes called ample analytic bundles. A weakly-positive bundle over a space $  X $
 +
naturally defines an imbedding of $  X $
 +
into a Grassmann manifold and thus into a projective space.
  
The concepts of a positive, a negative, a weakly-positive, and a weakly-negative bundle are naturally extended also to the case of a linear space over a complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073980/p07398078.png" /> (see [[Vector bundle, analytic|Vector bundle, analytic]]).
+
The concepts of a positive, a negative, a weakly-positive, and a weakly-negative bundle are naturally extended also to the case of a linear space over a complex space $  X $(
 +
see [[Vector bundle, analytic|Vector bundle, analytic]]).
  
 
See also [[Negative vector bundle|Negative vector bundle]].
 
See also [[Negative vector bundle|Negative vector bundle]].

Latest revision as of 08:07, 6 June 2020


A generalization of the concept of a divisor of positive degree on a Riemann surface. A holomorphic vector bundle $ E $ over a complex space $ X $ is called positive (denoted by $ E > 0 $) if on $ E $ there exists a Hermitian metric $ h $ such that the function

$$ v \mapsto - h( v, v) $$

on $ E $ is strictly pseudo-convex outside the zero section. If $ X $ is a manifold, the condition for positivity is expressed in terms of the curvature of the metric $ h $. That is, the curvature form of the metric $ h $ in the bundle $ E $ corresponds to a Hermitian quadratic form $ \Omega $ on $ X $ with values in the bundle $ \mathop{\rm Herm} ( E) $ of Hermitian endomorphisms of the bundle $ E $. The positivity condition is equivalent to $ \Omega _ {x} ( u) $ being a positive-definite operator on $ E _ {x} $ for any $ x \in X $ and any non-zero $ u \in T _ {X,x } $.

If $ E $ is a complex line bundle over a manifold $ X $, the condition for positivity is equivalent to that of positive definiteness of the matrix

$$ \left \| - \frac{\partial ^ {2} \mathop{\rm log} h }{\partial z _ \alpha \partial {\overline{z}\; } _ \beta } \right \| , $$

where $ z _ {1} \dots z _ {n} $ are local coordinates in $ X $ and $ h > 0 $ is a function that defines the Hermitian metric for the local trivialization of the bundle. If $ X $ is compact, then the complex line bundle $ E $ over $ X $ is positive if and only if the Chern class $ c _ {1} ( E) $ contains a closed form of the type

$$ i \sum _ {\alpha , \beta } \phi _ {\alpha \beta } dz _ \alpha d {\overline{z}\; } _ \beta , $$

where $ \| \phi _ {\alpha \beta } \| $ is a positive-definite Hermitian matrix. In particular, if $ X $ is a Riemann surface, then the bundle over $ X $ defined by a divisor of degree $ d $ is positive if and only if $ d > 0 $. If $ E $ is a bundle of rank $ > 1 $ over a manifold $ X $ of dimension $ > 1 $, one can consider also the following narrower class of positive bundles: A bundle $ E \rightarrow X $ is called positive in the sense of Nakano if there exists on $ E $ a Hermitian metric $ h $ such that the Hermitian quadratic form $ H $ on the bundle $ E \otimes T _ {X} $ as defined by the formula

$$ H _ {x} ( v \otimes u) = h _ {x} ( v, \Omega _ {x} ( u) v), $$

where $ x \in X $, $ v \in E _ {x} $ and $ u \in T _ {X,x } $, is positive definite. Examples: the tangent bundle $ TP _ {n} $ of the projective space $ P ^ {n} $ is positive, but for $ n > 1 $ it is not positive in the sense of Nakano; the complex line bundle over $ P ^ {n} $ defined by a hyperplane is positive.

Any quotient bundle of a positive vector bundle is positive. If $ E ^ \prime $ and $ E ^ {\prime\prime} $ are positive bundles (in the sense of Nakano), then $ E ^ \prime \oplus E ^ {\prime\prime} $ and $ E ^ \prime \otimes E ^ {\prime\prime} $ are positive (in the sense of Nakano).

The concept of a positive bundle was introduced in connection with Kodaira's vanishing theorem (cf. Kodaira theorem) for complex line bundles, and it was then extended to any bundle. Somewhat later, in relation to the existence of an imbedding in a projective space, the concepts of weakly-positive and weakly-negative bundles were introduced.

A holomorphic vector bundle $ E $ over a compact complex space $ X $ is called weakly negative if its zero section has a strictly pseudo-convex neighbourhood in $ E $, i.e. if it is an exceptional analytic set. A bundle $ E $ is called weakly positive if the dual bundle $ E ^ {*} $ is weakly negative. If $ X $ is a Riemann surface, the concepts of a weakly-positive bundle and a positive bundle coincide [5]. In the general case, positivity implies weak positivity; no examples are known at present (1983) of weakly positive but non-positive bundles.

Weak positivity of a bundle $ E \rightarrow X $ is equivalent to each of the following properties: For any coherent analytic sheaf $ {\mathcal F} $ on $ X $ there exists an $ m _ {0} > 0 $ such that $ {\mathcal F} \otimes S ^ {m} {\mathcal E} $ for $ m \geq m _ {0} $ is generated by global sections; for any analytic sheaf $ {\mathcal F} $ on $ X $ there exists an $ m \geq 0 $ such that

$$ H ^ {q} ( X, {\mathcal F} \otimes S ^ {m} {\mathcal E}) = 0 $$

for all $ q \geq 1 $[3], [4]. By $ {\mathcal E} $ is meant the sheaf of germs of holomorphic sections of the bundle $ E $. Weakly-positive bundles are therefore analogous to the ample sheafs (cf. Ample sheaf) from algebraic geometry and are sometimes called ample analytic bundles. A weakly-positive bundle over a space $ X $ naturally defines an imbedding of $ X $ into a Grassmann manifold and thus into a projective space.

The concepts of a positive, a negative, a weakly-positive, and a weakly-negative bundle are naturally extended also to the case of a linear space over a complex space $ X $( see Vector bundle, analytic).

See also Negative vector bundle.

References

[1] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004
[2] S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) MR0533884 Zbl 0444.32004
[3] A.L. Onischik, "Pseudoconvexity in the theory of complex spaces" J. Soviet Math. , 14 : 4 (1980) pp. 1363–1406 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 15 (1977) pp. 93–171
[4] M. Schneider, "Familien negativer Vektorraumbündel und 1-konvexe Abbildungen" Abh. Math. Sem. Univ. Hamburg , 47 (1978) pp. 150–170 MR0492393 Zbl 0391.32011
[5] H. Umemura, "Some results in the theory of vector bundles" Nagoya Math. J. , 52 (1973) pp. 97–128 MR0337968 Zbl 0271.14005 Zbl 0253.14003
How to Cite This Entry:
Positive vector bundle. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_vector_bundle&oldid=23928
This article was adapted from an original article by r group','../u/u095350.htm','Unitary transformation','../u/u095590.htm','Vector bundle, analytic','../v/v096400.htm')" style="background-color:yellow;">A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article