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Here is a question about construction of moduli space of Higgs bundles or Hitchin pairs. Let's say we are constructing the moduli space of stable Hitchin pairs or Higgs bundles (doesn't matter at this point), E-->E\otimes K over a smooth curve, a surface, etc with the condition that E is a stable sheaf. Is this construction done somewhere? In particular I am concerned about the properness of this moduli space (the stable E's cut an open subscheme inside their projective Quot scheme). So before even discussing the Higgs stability, we are dealing with an open scheme of E's. Is there a proof somewhere which uses valuative criteria etc, to prove the properness of this particular moduli space?---It would be great if someone else already did this (?). Thank you.

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    $\begingroup$ What if $(E, E\to E\otimes K)$ is stable as a Higgs bundle but $E$ alone is unstable? $\endgroup$ Mar 4 '17 at 19:27
  • $\begingroup$ Well that is ok, and such moduli spaces have been constructed before. in normal situation; we first construct the Quot scheme of E's, then there is a scheme that parameterizes the morphisms E-->E\otimes K , then we cut the Higgs-stable locus inside that latter scheme and mod out by Auto's (of the pair). Simpson I believe constructed these moduli spaces of HiItchin pairs, as moduli space of some modules over a dg algebra (he calls them \Lambda modules ? ) but I dont think he required the E's to be stable in the first place, and that is what my main question is about. $\endgroup$
    – Artan
    Mar 4 '17 at 21:02
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    $\begingroup$ What I meant is that it might (and in fact does, if I remember correctly) happen that a family $(E_t, psi_t:E_t\to E_t\otimes K)$ of (Higgs-)stable Higgs bundles has $E_t$ stable for $t\neq 0$ while $E_0$ is not stable. This seems to imply that the moduli space your question is about is non-proper. $\endgroup$ Mar 4 '17 at 21:35
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Even if we restrict to Higgs bundles where $E$ is stable as a vector bundle, the moduli space is not proper. If you take $(E,\sigma)$ with $\sigma \neq 0$, then $$\lim_{t \to \infty} (E,t\sigma)$$ does not exist.

Another way to see this is that there are nonconstant functions on the moduli space of Higgs bundles (think of the Hitchin fibration), which would be impossible if the moduli space were proper.

One way to think about this failure of this properness is that the moduli space of stable Higgs bundles is isomorphic to a moduli space of stable torsion sheaves on the total space of the cotangent bundle of your base $X$. The moduli space is not proper, since the total space isn't proper, but there's a fairly natural compactification coming from taking torsion sheaves on the projective completion of the total space of the cotangent bundle. We can think of this as allowing the spectral varieties of the Higgs bundles to go to infinity.

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  • $\begingroup$ Thanks, so in the case of torsion sheaves on total space of the canonical of X, there is a torus acting on the moduli space---so here stable torsion shaves are in 1-1 correspondence with stable pairs downstairs, which is induced by the action of the torus which scales the fibers of K, and torus fixed locus of the moduli space is proper, as long as the base X is projective. I wonder if we restrict to stable sheaves E in this situation, whether we could argue that torus-fixed locus of the moduli space of E-->E\otimes K_X (where E is stable ) is proper (?). $\endgroup$
    – Artan
    Mar 7 '17 at 18:18
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    $\begingroup$ The torus fixed locus is going to correspond to the locus where the map $E \to E \otimes K_X$ is the zero map. I would expect that this doesn't have to be proper in general, but I haven't thought hard about it. $\endgroup$
    – user36254
    Mar 8 '17 at 19:53

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