Bridgeland stability and Gieseker stability I: Limit stability

We study the large volume limit of Bridgeland stability conditions on smooth projective varieties. For each fixed ray in a suitable angular sector, we prove that an object of $\mathrm{D}^b(X)$ is Bridgeland (semi)stable in the corresponding hearts for all sufficiently large radii if and only if it is a pure $H$-Gieseker (semi)stable sheaf. In this sector, we identify the category of coherent sheaves as the limiting heart in the sense of eventual membership. We also identify a perverse coherent limiting heart in the opposite angular sector. The key ingredients are uniform HN-phase estimates for bounded families of pure sheaves, coefficient bounds for the coherent images of potential destabilizers, and a criterion forcing narrow exact sequences in the varying hearts to be bounded saturated coherent sequences.