Relative vanishing theorems for Q\mathbf{Q}-schemes

We prove the relative Grauert-Riemenschneider vanishing, Kawamata-Viehweg vanishing, and Koll\'ar injectivity theorems for $\mathbf{Q}$-schemes, solving conjectures of Boutot and Kawakita. Our proof uses Grothendieck's limit theorem for sheaf cohomology and Zariski-Riemann spaces. As an application, we extend Boutot's theorem to the case of locally quasi-excellent $\mathbf{Q}$-algebras by showing that if $R \to R'$ is a cyclically pure homomorphism of locally quasi-excellent $\mathbf{Q}$-algebras, and $R'$ has rational singularities, then $R$ has rational singularities. This solves a conjecture of Boutot and answers a question of Schoutens in the locally quasi-excellent case.Comment: 31 pages. Comments welcome! v2: Removed excellence hypotheses, added references, other change

Dilated Floor Functions That Commute

We determine all pairs of real numbers $(\alpha, \beta)$ such that the dilated floor functions $\lfloor \alpha x\rfloor$ and $\lfloor \beta x\rfloor$ commute under composition, i.e., such that $\lfloor \alpha \lfloor \beta x\rfloor\rfloor = \lfloor \beta \lfloor \alpha x\rfloor\rfloor$ holds for all real $x$.Comment: 6 pages, to appear in Amer. Math. Monthl

Pure subrings of Du Bois singularities are Du Bois singularities

Let $R \to S$ be a cyclically pure map of rings essentially of finite type over the complex numbers $\mathbb{C}$. In this paper, we show that if $S$ has Du Bois singularities, then $R$ has Du Bois singularities. Our result is new even when $R \to S$ is faithfully flat. As a consequence, we show that under the same hypotheses on $R \to S$, if $S$ has log canonical type singularities and $K_R$ is Cartier, then $R$ has log canonical singularities.Comment: 7 pages; comments welcome. v2: Added part (ii) of Corollary B, other small change

Permanence properties of FF-injectivity

We prove that $F$-injectivity localizes, descends under faithfully flat homomorphisms, and ascends under flat homomorphisms with Cohen-Macaulay and geometrically $F$-injective fibers, all for arbitrary Noetherian rings of prime characteristic. As a consequence, we show that the $F$-injective locus is open on most rings arising in arithmetic and geometry. Furthermore, we prove that over an algebraically closed field of positive characteristic, generic projection hypersurfaces associated to normal projective varieties are weakly normal, and generic projection hypersurfaces associated to suitably embedded smooth projective varieties of low dimension are even $F$-pure, and hence $F$-injective. The former result proves a conjecture of Bombieri and Andreotti-Holm, and the latter result is the positive characteristic analogue of a theorem of Doherty.Comment: 38 pages; comments welcome! v2: added Theorem 6.6, fixed Lemma A.2, more transparent proof of Lemma 4.5, other small additions and change

The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero

We establish the relative minimal model program with scaling for projective morphisms of quasi-excellent algebraic spaces admitting dualizing complexes, formal schemes admitting dualizing complexes, semianalytic germs of complex analytic spaces, rigid analytic spaces, and Berkovich spaces, all in equal characteristic zero. To do so, we prove finite generation of relative adjoint rings associated to projective morphisms of such spaces using the strategy of Cascini and Lazi\'c and the generalization of the Kawamata-Viehweg vanishing theorem to the scheme setting recently established by the second author. To prove these results uniformly, we prove GAGA theorems for Grothendieck duality and dualizing complexes to reduce to the algebraic case.Comment: 96 pages. Comments welcom