In this talk I will discuss a conjecture on vanishing theorems for Fano's in positive characteristic.

I will report on what I know about the progress on the problem, including the solution in dimension two (joint work with J. Lacini and F. Bernasconi). Finally, I will discuss the relationship (due to C. Hacon and J. Witaszek) between this vanishing theorem and the depth of klt singularities and explain how similar ideas can be used to construct a three-dimensional log-canonical singularity which contradicts a theorem of Kollár over the complex numbers in every positive characteristic. This example is part of a joint work in progress with F. Bernasconi and Z. Patakfalvi. Speaker(s): Emelie Arvidsson (IAS)