Vanishing theorems in algebraic geometry

Abstract: Vanishing theorems in complex 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 this story, I will survey some recent and ongoing work giving analogous results in mixed characteristic as well as applications to birational geometry; the key input now comes from p-adic Hodge theory.