Given an arithmetic variety $f: X \rightarrow Spec(\mathbb{Z})$.

Is there a notion of boundedness for families of sheaves on $X$?

I only found the notion for families on the fibers of $f$. But i am interested in sheaves defined on $X$.

All definitions / theorems i found only work when $X$ is defined over some field $k$, where one has the Hilbert polynomial, slope etc, which we don't have in this case. Is there some substitute for these terms?

Or are there even results about moduli spaces of sheaves on arithmetic varieties?

Edit: According to http://arxiv.org/abs/math/0612268 there is a notion of arithmetic (semi)stability. One even has a Harder Narasimhan filtration. Can one define the notion of boundedness in the Arakelov setting? Are there any results on moduli spaces of vector bundles in Arakelov theory?


1 Answer 1


When doing moduli theory over $\mathbb Z$, or another base scheme, one works with sheaves that are flat over the base; this implies that all the discrete invariants, such as the Hilbert polynomial, are constant in the fibers. Stability is defined fiber by fiber; i.e., a sheaf is (semi)stable when it it (semi)stable on all the fibers. The Quot schemes of sheaves with fixed Hilbert polynomial are defined and projective over $\mathbb Z$; then the standard boundedness results all generalize. Thus one obtains stacks of stable, or semistable, bundles, which are defined over $\mathbb Z$. When the existence of (quasi)projective moduli spaces is obtained via GIT, this also works over $\mathbb Z$ (a result of Seshadri, see Geometric reductivity over arbitrary base, Adv. Math. 26 (1977), 225–274).

There is an issue of when the fiber of one of these moduli spaces over a prime $p$ is the moduli space of the corresponding sheaves on the fiber of $X$ over $p$; this is not automatic, because the formation of moduli spaces in positive or mixed characteristic does not, in general, commute with non-flat base chage. If this comes up, it has to be analyzed case by case.

I hope this is what you what. If the question is the construction of a space whose points corresponds to global sheaves on $X$ with metric at infinity, defining stability by some kind of Arakelov-theoretic Hilbert polynomial, then I don't have a clue.

  • $\begingroup$ The Arakelov theory seems to be the thing i've been searching for. Thanks for bringing this to my attention! I included it in my first question. $\endgroup$
    – TonyS
    Apr 10, 2010 at 13:00

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