# Positive vector bundle

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 . 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 $q \geq 1$, . 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).