[Paper Review] Relative vanishing theorems for $\mathbf{Q}$-schemes

Takumi Murayama|arXiv (Cornell University)|Jan 25, 2021

Algebraic Geometry and Number Theory67 references4 citations

TL;DR

This paper establishes relative vanishing and injectivity theorems for $$\mathbf{Q}$-schemes and related analytic spaces in equal characteristic zero, using Zariski–Riemann spaces and the Grothendieck limit theorem. It resolves long-standing conjectures of Boutot and Kawakita by proving Kawamata–Viehweg vanishing and Kollár injectivity for proper morphisms of excellent schemes with dualizing complexes.

ABSTRACT

We prove the relative Grauert-Riemenschneider vanishing, Kawamata-Viehweg vanishing, and Kollár injectivity theorems for proper morphisms of schemes of equal characteristic zero, solving conjectures of Boutot and Kawakita. Our proof uses the Grothendieck limit theorem for sheaf cohomology and Zariski-Riemann spaces. We also show these vanishing and injectivity theorems hold for locally Moishezon (resp. projective) morphisms of quasi-excellent algebraic spaces admitting dualizing complexes and semianalytic germs of complex analytic spaces (resp. quasi-excellent formal schemes admitting dualizing complexes, rigid analytic spaces, Berkovich spaces, and adic spaces locally of weakly finite type over a field), all in equal characteristic zero. We give many applications of our vanishing results. For example, we extend Boutot's theorem to all Noetherian $\mathbf{Q}$-algebras by showing that if $R o R'$ is a cyclically pure map of $\mathbf{Q}$-algebras and $R'$ is pseudo-rational, then $R$ is pseudo-rational. This solves a conjecture of Boutot and affirmatively answers a question of Schoutens. The proof of this Boutot-type result uses a new characterization of pseudo-rationality and rational singularities using Zariski-Riemann spaces. This characterization is also used in the proofs of our vanishing and injectivity theorems and is of independent interest.

Motivation & Objective

• To extend classical vanishing theorems—such as Kawamata–Viehweg and Kollár injectivity—to proper morphisms of Noetherian $$\mathbf{Q}$-schemes and broader analytic categories.
• To resolve conjectures by Boutot and Kawakita concerning the validity of relative vanishing theorems in equal characteristic zero beyond smooth or projective varieties.
• To establish a new characterization of rational singularities and pseudo-rationality via Zariski–Riemann spaces, independent of resolution of singularities.
• To extend these results to locally Moishezon algebraic spaces, formal schemes, rigid and Berkovich spaces, and complex analytic germs, all in equal characteristic zero.
• To apply the vanishing theorems to prove new results on rational singularities, Gorenstein properties of complete local UFDs, and pseudo-rationality under cyclically pure maps.

Proposed method

• Utilizes the Grothendieck limit theorem for sheaf cohomology to analyze limits of cohomology groups along inverse systems of schemes.
• Employs Zariski–Riemann spaces as a tool to characterize rational singularities and pseudo-rationality via limits of resolutions.
• Applies the theory of dualizing complexes on excellent schemes to define canonical sheaves and adjunctions in the relative setting.
• Reduces the general case to the projective case via Chow’s lemma and log resolutions in various geometric categories (algebraic spaces, analytic germs, rigid, Berkovich, adic spaces).
• Constructs a relative log resolution $$\tilde{X}\to X$ for klt pairs and uses the projection formula to transfer vanishing from $$\tilde{X}$ to $$X$.
• Uses the Leray spectral sequence to relate cohomology of $$f_*$ to that of $$f\circ\pi\circ\mu_*$, leveraging vanishing on the resolution to deduce vanishing on the original scheme.

Experimental results

Research questions

• RQ1 Does the Kawamata–Viehweg vanishing theorem hold for proper morphisms of Noetherian $$\mathbf{Q}$-schemes with dualizing complexes?
• RQ2 Can Kollár’s injectivity theorem be extended to klt pairs over general excellent schemes in equal characteristic zero?
• RQ3 Is there a characterization of rational singularities and pseudo-rationality that avoids resolution of singularities and uses Zariski–Riemann spaces?
• RQ4 Do relative vanishing theorems extend to non-Archimedean and complex analytic settings such as rigid, Berkovich, or formal schemes?
• RQ5 Does cyclically pure surjection preserve pseudo-rationality for $$\mathbf{Q}$-algebras, as conjectured by Boutot and Schoutens?

Key findings

• The relative Kawamata–Viehweg vanishing theorem holds for proper maximally dominating morphisms of Noetherian $$\mathbf{Q}$-schemes with dualizing complexes, under klt or simple normal crossing assumptions.
• Kollár’s injectivity theorem extends to klt pairs over excellent schemes in equal characteristic zero, with injectivity of $R^if_*\mathcal{O}_X(N) \to R^if_*\mathcal{O}_X(N+D)$ for $f$-semi-ample $M$.
• A new characterization of rational singularities and pseudo-rationality is established using limits over Zariski–Riemann spaces, independent of resolution.
• The vanishing theorems are extended to locally Moishezon algebraic spaces, quasi-excellent formal schemes, and semianalytic germs of complex analytic spaces.
• The paper proves that if $R \to R'$ is a cyclically pure map of $$\mathbf{Q}$-algebras and $R'$ is pseudo-rational, then $R$ is pseudo-rational, affirming a conjecture of Boutot and a question of Schoutens.
• The results are applied to show that complete local UFDs of dimension $\leq 4$ with residue field $\mathbb{C}$ are Gorenstein, under mild assumptions on the dualizing complex.