First I will talk about my joint work with Chris Lazda on how to use some elementary properties of the logarithmic de Rham-Witt complex to prove that a rational line bundle on the special fibre of a proper, semistable scheme over a power series ring k[[t]] in characteristic p lifts to the total space if and only if its first Chern class does. This generalises a result of Morrow in the smooth case, and provides an equicharacteristic analogue of a result of Yamashita. I will also explain an application to the Tate conjecture for certain Drinfeld-Stuhler modular varieties; the latter is joint work in progress with J