Degree (vector bundle)

The degree of a vector bundle on a projective algebraic curve is a relatively coarse, integer invariant . It is closely related to the Euler characteristic of the vector bundle. Trivial vector bundles are of degree 0.

definition

The degree of a straight line with a divisor is defined as the degree of . If a divisor, then its degree is simply the whole number . The degree of a vector bundle is the degree of its determinant bundle . ${\displaystyle \deg {\mathcal {L}}}$ ${\displaystyle {\mathcal {L}}\cong {\mathcal {O}}(D)}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle \textstyle D=\sum n_{P}P}$ ${\displaystyle \textstyle \sum n_{P}}$ ${\displaystyle D}$ ${\displaystyle \textstyle \det E=\bigwedge ^{{\rm {rk}}E}E}$

A meromorphic section in a bundle of straight lines has zeros and poles , each of which occurs with a certain order (multiplicity) . The total sum of these orders, whereby the polar orders have to be counted negatively, is independent of the meromorphic section itself, and is precisely the degree of the bundle. ${\displaystyle \neq 0}$

properties

• Additivity: is

${\displaystyle 0\longrightarrow {\mathcal {E}}'\longrightarrow {\mathcal {E}}\longrightarrow {\mathcal {E}}''\longrightarrow 0}$

is a short exact sequence of vector bundles

${\displaystyle \deg {\mathcal {E}}=\deg {\mathcal {E}}'+\deg {\mathcal {E}}''.}$

• The following applies for two vector bundles ${\displaystyle {\mathcal {E}},{\mathcal {F}}}$

${\displaystyle \deg({\mathcal {E}}\otimes {\mathcal {F}})=\mathrm {rk} \,{\mathcal {F}}\cdot \deg {\mathcal {E}}+\mathrm {rk} \,{\mathcal {E}}\cdot \deg {\mathcal {F}}.}$

• Riemann-Roch theorem : For a vector bundle on a smooth curve of gender : ${\displaystyle {\mathcal {E}}}$ ${\displaystyle g}$

${\displaystyle \chi ({\mathcal {E}})=\deg {\mathcal {E}}+\mathrm {rk} \,{\mathcal {E}}\cdot (1-g).}$

The degree on higher dimensional varieties

On a smooth (or at least normal) projective variety of any dimension , a vector bundle (and even more generally a torsion-free sheaf) can also be assigned a degree, which, however, depends on a fixed, very amplified divisor . In this situation one sets (using the intersection theory ) ${\displaystyle X}$ ${\displaystyle E}$ ${\displaystyle H}$

${\displaystyle \deg \,E:=\det(E)\,.\,H^{\rm {dim(X)-1}}}$

So take the -fold self-intersection of the amplified divisor, which gives a curve, and consider the restriction of the bundle to this curve. This degree has properties similar to the degree defined on a curve. ${\displaystyle ({\rm {dim(X)-1)}}}$

This definition is simplified for a vector bundle in a projective space . One has with a uniquely determined whole number , which is called the degree. ${\displaystyle \det E=O(n)}$ ${\displaystyle n}$

The slope of a vector bundle

For a given vector bundle , one defines (for the first time by David Mumford ) the slope (in German: the inclination, but this is not common) as ${\displaystyle E}$

${\displaystyle \mu (E):={\frac {\deg(E)}{{\rm {rk}}(E)}}\,.}$

This is the starting point of the theory of (semi) stable vector bundles .