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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]]  $  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

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

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|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|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|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|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  $  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

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

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  $  X $(
see [[Vector bundle, analytic|Vector bundle, analytic]]).

See also [[Negative vector bundle|Negative vector bundle]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) {{MR|0608414}} {{ZBL|0435.32004}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) {{MR|0533884}} {{ZBL|0444.32004}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> M. Schneider, "Familien negativer Vektorraumbündel und 1-konvexe Abbildungen" ''Abh. Math. Sem. Univ. Hamburg'' , '''47''' (1978) pp. 150–170 {{MR|0492393}} {{ZBL|0391.32011}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> H. Umemura, "Some results in the theory of vector bundles" ''Nagoya Math. J.'' , '''52''' (1973) pp. 97–128 {{MR|0337968}} {{ZBL|0271.14005}} {{ZBL|0253.14003}} </TD></TR></table>