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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:

  1. 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?

  1. 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://mathoverflow.net/questions/371022/on-definition-of-stable-vector-higgs-bundle?noredirect=1#comment937997_371022

-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?

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    $\begingroup$ For 2), you might want to have a look at 4.C. in Huybrechts-Lehn's "The geometry of moduli spaces of sheaves". Also have a look at these papers arxiv.org/abs/1505.07091 and arxiv.org/abs/alg-geom/9402008 $\endgroup$
    – Bananeen
    Sep 6, 2020 at 16:09
  • $\begingroup$ @Bananeen, I didn't look into Huybrechts-Lehn's yet, although the paper of Bertram,Martinez seems to deal completely with the Gieseker stability. Tthe slope-stability is a priori different, and I have never met any kind of this wall-crossing business for slope-stable vector bundles (this is what 2) is about, in fact) $\endgroup$
    –  V. Rogov
    Sep 7, 2020 at 10:35
  • $\begingroup$ Yeah, and Dolgachev and Hu is great, although this is a general theory; I wonder if there is something specifically about moduli spaces of vector and/or Higgs bundles. Still thank you! $\endgroup$
    –  V. Rogov
    Sep 7, 2020 at 10:37
  • $\begingroup$ Ah, ok, I am not that familiar with slope-stability as such besides the curve case. Probably then you should look at gauge-theoretic literature as well. The wall and chamber structure when you vary the Kahler class should be a quite general phenomenon, since the spaces are still constructed as the GIT quotient. For an elementary treatment, you can also take a look at section 1.6 in Schmitt's "Geometric invariant theory and decorated principal bundles" (probably you know this already at a general level, but just in case). $\endgroup$
    – Bananeen
    Sep 7, 2020 at 12:45

1 Answer 1

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Here is a simple example for question 1a.

Take a point in the projective plane with ideal sheaf $m$. One has the Koszul resolution $0\to O(-2)\to O(-1)\oplus O(-1)\to m\to 0\to 0$. Take a push out of $O(-2)\to O$ for a general such map and then we have an exact sequence $0\to O\to E\to m\to 0$ with $E$ a rank two vector bundle with first chern class zero and thus $\mu(E)=0$. It is not stable, since it has $O$ as a subsheaf which has $\mu=0$. Easy to check that it does not have a destabilizing sub bundle.

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  • $\begingroup$ Does that agree with the discussion on p. 73 of Le Potier's "Lectures on vector bundles", where it is shown that a destabilizing subsheaf is always contained in a destabilizing subbundle? $\endgroup$
    – Bananeen
    Sep 6, 2020 at 16:03
  • $\begingroup$ @Bananeen I do not have Le Potier with me right now, but I suspect he means it for curves, where it is correct. For higher dimensions, this is no longer true as my example shows. $\endgroup$
    – Mohan
    Sep 6, 2020 at 17:05
  • $\begingroup$ I see, this means my intuition has been wrong for years. Great example, thanks! $\endgroup$
    – Bananeen
    Sep 6, 2020 at 17:27
  • $\begingroup$ Yeah, Le Potier defenitely means this for curves, and due to this for curves the two notions coincide, I forgot to mention it in my question $\endgroup$
    –  V. Rogov
    Sep 7, 2020 at 10:26
  • $\begingroup$ @Mohan thank you, that's a good example! Although, it seems to me that $E$ is not stable, but still semi-stable, so I'd ask if this can also be violated? $\endgroup$
    –  V. Rogov
    Sep 7, 2020 at 10:29

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