Geometric models for the algebraic hearts in the derived category of a gentle algebra

We give a geometric model for any algebraic heart in the derived category of a gentle algebra, which is equivalent to the module category of some gentle algebra. To do this, we deform the geometric model for the module category of a gentle algebra given in [BC21], and then embed it into the geometric model of the derived category given in [OPS18], in the sense that each so-called zigzag curve on the surface represents an indecomposable module as well as the minimal projective resolution of this module. A key point of this embedding is to give a geometric explanation of the duality between the simple modules and the projective modules. Such a blend of two geometric models provides us with a handy way to describe the homological properties of a module within the framework of the derived category. In particular, we realize any higher Yoneda-extension as a polygon on the surface, and realize the Yoneda-product as gluing of these polygons. As an application, we realize any algebraic heart in the derived category of a gentle algebra on the marked surface.