Vanishing theorems for Zariski–Riemann spaces in equal characteristic zero and applications to rational singularities
Algebraic Geometry Seminar
Let S be a regular ring. By work of Hochster-Roberts, Boutot, Smith, Hochster-Huneke, Schoutens, and Heitmann-Ma, we know that every pure subring of S is pseudo-rational, whence Cohen-Macaulay. This applies for example to rings of invariants of linearly reductive groups. For pure maps R -> S of rings essentially of finite type over fields of characteristic zero, Boutot's result is even stronger: If S has rational singularities, then R has rational singularities. We show that Boutot's theorem holds more generally if R and S are arbitrary Noetherian Q-algebras, which gives new proofs of the theorems of Hochster-Huneke and Schoutens in equal characteristic zero. This solves a conjecture of Boutot and answers a question of Schoutens.
In this talk, I will discuss my generalization of Boutot’s theorem and the key new ingredient: a Kodaira-type vanishing theorem for Zariski–Riemann spaces of Noetherian schemes of equal characteristic zero. My vanishing theorem has many applications. For example, it implies vanishing theorems and the existence (proved jointly with Shiji Lyu) of the relative minimal model program with scaling for algebraic spaces, formal schemes, and both complex and non-Archimedean analytic spaces.