Bridgeland stability for restricted Kahler moduli?

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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
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