Stable vector bundle

In mathematics , stable and semi- stable vector bundles are a concept of the geometric invariant theory (in its modern formulation, which goes back to Mumford ).

Definitions

The slope of a vector bundle on a smooth projective curve is the quotient of degree and rank of . ${\ displaystyle \ mu (E)}$ ${\ displaystyle E}$ ${\ displaystyle {\ tfrac {deg (E)} {rk (E)}}}$ ${\ displaystyle E}$

A vector bundle is called stable if for every non-trivial sub-bundles applies: . is called semi-stable if the weaker condition is met. is called polystable if it is the direct sum of stable bundles. Line bundles, that is vector bundles of rank one, are always stable. ${\ displaystyle E}$ ${\ displaystyle F \ subset E}$ ${\ displaystyle \ mu (F) <\ mu (E)}$ ${\ displaystyle E}$ ${\ displaystyle \ mu (F) \ leq \ mu (E)}$ ${\ displaystyle E}$

Equivalent to this, a vector bundle is (semi-) stable if the following applies to every nontrivial quotient of : (or ). ${\ displaystyle E}$ ${\ displaystyle Q}$ ${\ displaystyle E}$ ${\ displaystyle \ mu (Q)> \ mu (E)}$ ${\ displaystyle \ mu (Q) \ geq \ mu (E)}$

This term comes from David Mumford and is crucial for the construction of modular rooms . Namely, you cannot parameterize all vector bundles by a geometric object, but only the (semi) stable ones. This construction generalizes the construction of the Jacobian variety of a curve for greater rank .

Examples

On the projective straight line , only the straight line bundles are stable, vector bundles of the form for integers and are semistable . This is based on Grothendieck's theorem , that every vector bundle on the projective straight line is the direct sum of straight line bundles, and every straight line bundle has the form with an integer . ${\ displaystyle P}$ ${\ displaystyle O (n) ^ {r}}$ ${\ displaystyle n}$ ${\ displaystyle r \ geq 0}$ ${\ displaystyle O (n)}$ ${\ displaystyle n}$

On an elliptic curve , the semi-stable vector bundles are direct sums of indivisible vector bundles of the same slope. According to Atiyah's classification, the indecomposable vector bundles are given by . Here L denotes a straight line bundle.

{\ displaystyle F_ {r, d} \ otimes L}

For curves of higher sex , the description of the semi-stable vector bundles is much more difficult.

A holomorphic vector bundle over a Riemann surface is semistable if it is a flat bundle with a unitary holonomy representation , it is stable if and only if is irreducible . The generalization of this fact to arbitrary (not necessarily unitary) representations leads to the theory of the Higgs bundle . ${\ displaystyle \ mathbb {C} ^ {n}}$ ${\ displaystyle \ Sigma}$ ${\ displaystyle \ rho \ colon \ pi _ {1} \ Sigma \ to U (n)}$ ${\ displaystyle \ rho}$

properties

Are and semi-stable, and is , so is because the image should have slope on the one hand and slope on the other . ${\ displaystyle E}$ ${\ displaystyle F}$ ${\ displaystyle \ mu (E)> \ mu (F)}$ ${\ displaystyle Hom (E, F) = 0}$ ${\ displaystyle \ geq \ mu (E)}$ ${\ displaystyle \ leq \ mu (F)}$

Harder-Narasimhan filtration

If there is any vector bundle, it has a functionally descending filtration parameterized by rational numbers , so that the filtration quotients are semi-stable with an increase . It is won by one ${\ displaystyle E}$ ${\ displaystyle E}$ ${\ displaystyle E ^ {\ alpha}}$ ${\ displaystyle \ alpha}$

the largest semi-stable sub-bundle is considered (it is also the largest of those sub-bundles that have the maximum slope) ${\ displaystyle F}$
the quotient is ${\ displaystyle E / F}$

and repeat this process.