Bridgeland stability for restricted Kahler moduli?

Question

Let $X$ be a simply-connected, smooth, projective Calabi-Yau threefold. To my understanding, Bridgeland introduced stability conditions on triangulated categories to give a proper mathematical definition of the stringy Kahler moduli space (SKMS) from physics.

Conjecturally, the classical (complexified) Kahler cone $\mathcal{K}_{X}(\mathbb{C})$ of $X$ gives an open chart on the SKMS around the large-volume limit. Coordinates on $\mathcal{K}_{X}(\mathbb{C})$ are called Kahler moduli, and depending on the context, one may prefer to think of them as formal variables tracking degrees along effective curve classes in $X$, i.e. effective classes in $H_{2}(X, \mathbb{Z})$.

Classically, it makes sense to consider only certain Kahler moduli: this would be some sort of sub-cone, or collection of sub-cones, in $\mathcal{K}_{X}(\mathbb{C})$. For example, one setting I'm interested in is when we have a proper surjective map

$$f: X \to \mathbb{P}^{1}$$

whose generic fibers are Calabi-Yau surfaces. You have certain Kahler moduli tracking curve classes in the fibers of $f$, and other Kahler moduli tracking classes "transverse" to the fibers. One might want to focus on just fiber classes, or transverse classes.

So my question is: can one expect submanifolds of the Bridgeland stability manifold/SKMS which correspond to only specific Kahler moduli, as I've described above?

For example, in the case of fiber classes of $f$, one can define the Serre subcategory $Coh(f)_{0}$ of $Coh(X)$ whose objects are coherent sheaves on $X$ supported on the fibers of $f$. You then get a full triangulated subcategory $D^{b}(X)_{f} \subset D^{b}(X)$ consisting of objects whose cohomology sheaves lie in $Coh(f)_{0}$.

By applying the machinery of Bridgeland to $D^{b}(X)_{f}$ or some similar triangulated subcategory, can one find a submanifold of the stability manifold/SKMS corresponding to fiber classes of $f$?

This post imported from StackExchange MathOverflow at 2020-01-22 12:09 (UTC), posted by SE-user Benighted