Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as
$\mu(\mathcal{E}) :=\frac{\operatorname{deg}(\mathcal{E})}{\operatorname{rk} \mathcal{E}}$
where $\operatorname{deg}(\mathcal{E})$ is defined as $c_1(\mathcal{E}) \cdot \omega^{n-2}$ for your favourite ample (or even Kähler) class $\omega$.
The bundle is called stable if for every subsheaf $\mathcal{F} \subset \mathcal{E}$, one has $\mu(\mathcal{F}) <\mu( \mathcal{E})$.
These are my questions:
- Why do we require that $\mathcal{E}$ has no subsheaves with greater slope, rather then no subbundles? Is the difference sufficient and what is the motivation for this choice? More precisely,
1a) is there an explicit example of a non-stable vector bundle, which has no subbundles with bigger slope?
1b) is there a reasonable moduli space for holomorphic vector bundles having no destabilising subbundles? If yes, how far is it from the moduli space of stable vector bundles?
2)The definition of slope a priori depends on the choice of Kähler form. How much does the moduli space of stable vector bundles depends on this choice?
- I am also interested in the analogues of the questions 1a) and 1b) for the Higgs bundles. In this case we require that a Higgs bundle $(\mathcal{E}, \theta)$ has no Higgs subsheaves(?) $(\mathcal{F}, \theta|_{\mathcal{F}})$ with bigger slope.
UPD: -cross-list with https://math.stackexchange.com/questions/3816328/slope-stability-subsheaves-vs-subbundles
-In 1a) one can ask the same for semi-stability;
-On a curve every destabilizing subsheaf is contained in a subbundle, hence there is no difference indeed;
-For Gieseker-stability the answer for 2) is the theory of wall-crossings, but a have never seen a version of it for slope-stability (and for Higgs bundles as well). Is it exist?
