Vanishing Theorems in Algebraic Geometry

Vanishing theorems in complex algebraic geometry, such as the Kodaira vanishing theorem, play an important role in understanding the structure of complex algebraic varieties. Ultimately, they rely on critical input from Hodge theory. After recalling some aspects of this story, I will discuss analogous results in mixed characteristic as well as applications; the key input now comes from $p$-adic Hodge theory.