Question

In a fabulous paper (http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf) Bridgeland showed that locally finite stability conditions on a triangulated category form a complex manifold (possibly infinite dimensional or empty, but there are interesting examples). It is a complex manifold with one chart, where by a chart of M I mean a morphism from M to a complex vector space.

My question is, what is known about this complex manifold? How do people study this manifold and are there any known constraint on these manifold besides the one I mentioned above?

Answer 1

This is not an answer about the structure that the stability manifold can have in general. However, here is an example (a family of them actually) of such a manifold for an interesting scheme, The Hilbert Scheme of points in $\mathbb{P}^2$. (and stability conditions.) Intuitively, this will say that the stability manifold has the same information as the cone of effective divisors.

Here are more details.

Let $X=\mathbb{P}^{2[n]}$ be the Hilbert scheme of points in $\mathbb{P}^2$. The dimension of $X$ is $2n$ and a theorem by Forgaty claims that $X$ is smooth and $Pic(X)\otimes \mathbb{Q}$ has always rank $2$ (independent of the number of points). Consequently the cone of effective divisors ($Eff(X)$) is always a cone in the plane. It is not hard to prove that such a cone $Eff(X)$ has finitely many chambers(subcones) depending solely on the birrational geometry of $X$.

Keep in mind that cones in the plane ($X_1,X_2$) look like the shaded regions here. (ignore the circle $Y$.)

Now, here comes the birational geometry background. It is true that for each chamber in the cone $Eff(X)$, we can get a birational model $X_D$ (possibly equal to $X$ but in most cases $X_D\ne X$ and $X_D$ birrational to $X$). The seconds paragraph above says that we have finitely many birrational models of $X$. Now here is the question:

Do these birationational models $X_D$ correspond to ``vary'' the stability condition in the Hilbert scheme of points $X=\mathbb{P}^{2[n]}$? (meaning, varying a point in the stability manifold.)

Put it more precisely,

Is there a modular interpretation for the birrational models $X_D$ in terms of moduli spaces of Bridgeland semi-stable objects?

The answer is yes. Here is a paper by Arcara, Bertram, Coskun and Huizenga where you can find the details.

Roughly speaking then, this paper gives us understanding of the Bridgeland stability manifold (rather a one-dimensional section of it) as a cone in the plane where changing the stability condition (varing a point on the stability manifold) of the Hilbert points corresponds to undergo a birational transformation (and consequently getting a $X_D$.)