Geometry of Shimura varieties and positivity of automorphic line bundles on Unitary Shimura varieties

The study of coherent cohomology on (the special fiber of) Shimura varieties has various applications to arithmetic problems, such as congruences of automorphic forms, weight part of Serre's conjecture, and liftability of mod p automorphic forms. One of the basic problems is to prove certain automorphic line bundles are ample, which yields vanishing of coherent cohomology. In this talk, we start from the ampleness criterion of modular line bundles on Hilbert modular varieties, and then explain how to formulate a generalization to unitary Shimura varieties. Our final result works for split unitary Shimura varieties.