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Let $C$ be an algebraic curve. One of the easiest examples of stabilty functions is

$$Z:Coh(C)/ \{ 0 \} \rightarrow \overline{\mathbb{H}};\ \ \ \ Z(E):=-deg(E)+i\cdot rk(E).$$

This induces the classical $\mu$-stability on vector bundles of given rank on $C$. I wonder what happens if one modifies this, for instance by putting $-a\cdot deg(E)+bi\cdot rk(E)$ for appropriate constants $a,b$. Does the corresponding moduli space gets (birationally?) deformed in some way? or does it change completely?

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First : I don't know how to construct a structure of "moduli space" on the set of semi-stable objects for a stability condition which is no longer a GIT stability condition (precisely because then I have no GIT construction).

However, the question has also a sense for the set of semi-stable objects. The general picture is the following. When you slightly modify a generic stability condition, the set of semi-stable objects does not change. But there is some exceptionnal locus in the "space of stability condition" (in the sense of Bridgeland) such that when you pass it, the set of semi-stable object changes completely ("wall-crossing phenomena").

For a concrete example in the spirit example, one can consider Coh(P^1) and look at the various stability conditions of the form u deg(E) + v rk(E), u,v in the upper half plane plus the negative real half-line. In fact, as I don't remember the precise result, consider Rep(Q) the category of the representation of the quiver Q with two vertices linked by two arrows (in the same direction). In Rep(Q), we have U of vector dimension (1,0) and V of vector dimension (0,1). Consider stability condition on Rep(Q) of the form W -> u w1 + v w2 where (w1, w2) is the vector dimension of W. Then, one can show that all depend of the relative position of u and v in the upper half plane (more precisely : of the sign of the angle between u and v). In one case, there is a infinite number of semi-stable objects, in the other case, there is a finite number of semi-stable objects. The case of Coh(P^1) is almost the same (in fact Coh(P^1) and Rep(Q) have the same derived category) : there are two cases, one with infinite, the other with finite semi-stable objects.

I think that among projective curves, we have such a wall-crossing phenomena only for the projective line P^{1}. In higher genus, there is no such change (computation of the space of stablity condition for curves : Bridgeland, Macri, Okada).

Last comment : in higher dimension "standard" = GIT stability condition is not a stability condition in the sense of Bridgeland and I don't know how to answer the analogous of the question in this case.

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Without going over all of the details before I eat breakfast, I will go out on a limb and say the following: for $a$, $b$ close to 1, you will get an identical moduli space for the reason that stability is an open condition. In other words, you can change a GIT stability condition a little without changing which sheaves are stable.

In general, this kind of change to the stability condition goes under the heading of a "variation of GIT quotient." I think the relevant paper to look at is:

Geometric Invariant Theory and Flips by Michael Thaddeus

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  • $\begingroup$ Thank you for the hint! so, in the end, one can observe all this on the level of VGIT with no need of considering the derived category etc. did I get it right? $\endgroup$
    – IMeasy
    Aug 27, 2012 at 13:40
  • $\begingroup$ If I wasn't out of my depths before, certainly I am now. I do not understand how GIT stability and Bridgeland stability compare to one another. $\endgroup$
    – David Steinberg
    Aug 27, 2012 at 13:44
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    $\begingroup$ There won't be any exotic stability conditions for vector bundles on smooth curves. This has to do with the Picard group of the stable moduli space being isomorphic to Z. In other moduli problems, such as parabolic bundles, the flipped moduli space is Proj of the ring of sections of powers of a non-ample bundle. Varying the Bridgeland parameters as proposed by the questioner might replace stable bundles with their shifts in the derived category, or something like that, but there should be an obvious autoequivalence, like a shift, exchanging the old stable objects with the new ones. $\endgroup$
    – Michael Thaddeus
    Aug 29, 2012 at 14:19
  • $\begingroup$ I see. Is there a reference for the example of parabolic bundles that you bring ? $\endgroup$
    – IMeasy
    Aug 30, 2012 at 10:51

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