Symmetric differentials on singular surfaces and hyperbolicity

On a complex projective surface of general type, it is believed that outside a proper Zariski closed subset, the canonical degree of irreducible curves of fixed geometric genus is bounded. This holds, for example, when the cotangent bundle is big, i.e., there are many symmetric differentials. This is the case when the Chern numbers satisfy $c_1^2>c_2$, but there are other examples, such as the minimal resolutions of sufficiently nodal surfaces in P3. We review the notion of reflexive sheaves on singular surfaces and show how to use these sheaves to derive results about the presence of symmetric differentials on nodal surfaces (and their resolutions). These techniques also have a place in understanding rational curves on smooth surfaces. These curves can be contracted to obtain a singular surface, and inequalities involving invariants of the reflexive sheaves on the singular surface allow one to investigate the behavior of the canonical degree of the rational curves on the original surface. These techniques will be discussed together with specific calculations and possible extensions.