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

Shiji Lyu, Takumi Murayama

TL;DR

The paper proves the relative minimal model program with scaling for locally projective morphisms across several categories (excellent algebraic spaces, formal schemes, and analytic spaces) in equal characteristic zero, with extensions to positive/mixed characteristic in low dimensions. The core strategy reduces to the algebraic setting via new GAGA-type duality for Grothendieck duality and dualizing complexes, enabling uniform treatment across categories. Finite generation of relative adjoint rings is established using Cascini–Lazić’s approach in combination with Mur’s Kawamata–Viehweg vanishing, together with new Bertini-type results for relatively generated sheaves, to run and terminate the MMP with scaling. The work also derives gluing techniques over affinoid covers to patch local MMP steps and extends the framework to complex analytic spaces without shrinking the base, highlighting a broad unification of birational geometry tools across diverse spaces. These results illuminate the role of excellence, duality, and scalable strategies in extending the minimal model program beyond traditional quasi-projective varieties, with concrete new tools for analytic and non-Archimedean geometries.

Abstract

We establish the relative minimal model program with scaling for locally projective morphisms of quasi-excellent algebraic spaces admitting dualizing complexes, quasi-excellent formal schemes admitting dualizing complexes, semianalytic germs of complex analytic spaces, rigid analytic spaces, Berkovich spaces, and adic spaces locally of weakly finite type over a field, 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 Lazić 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. In addition, we apply our methods to establish the relative minimal model program with scaling for spaces of the form above in dimensions $\le 3$ in positive and mixed characteristic, and to show that one can run the relative minimal model program with scaling for complex analytic spaces without shrinking the base at each step.

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

TL;DR

The paper proves the relative minimal model program with scaling for locally projective morphisms across several categories (excellent algebraic spaces, formal schemes, and analytic spaces) in equal characteristic zero, with extensions to positive/mixed characteristic in low dimensions. The core strategy reduces to the algebraic setting via new GAGA-type duality for Grothendieck duality and dualizing complexes, enabling uniform treatment across categories. Finite generation of relative adjoint rings is established using Cascini–Lazić’s approach in combination with Mur’s Kawamata–Viehweg vanishing, together with new Bertini-type results for relatively generated sheaves, to run and terminate the MMP with scaling. The work also derives gluing techniques over affinoid covers to patch local MMP steps and extends the framework to complex analytic spaces without shrinking the base, highlighting a broad unification of birational geometry tools across diverse spaces. These results illuminate the role of excellence, duality, and scalable strategies in extending the minimal model program beyond traditional quasi-projective varieties, with concrete new tools for analytic and non-Archimedean geometries.

Abstract

We establish the relative minimal model program with scaling for locally projective morphisms of quasi-excellent algebraic spaces admitting dualizing complexes, quasi-excellent formal schemes admitting dualizing complexes, semianalytic germs of complex analytic spaces, rigid analytic spaces, Berkovich spaces, and adic spaces locally of weakly finite type over a field, 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 Lazić 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. In addition, we apply our methods to establish the relative minimal model program with scaling for spaces of the form above in dimensions in positive and mixed characteristic, and to show that one can run the relative minimal model program with scaling for complex analytic spaces without shrinking the base at each step.
Paper Structure (73 sections, 109 theorems, 360 equations)

This paper contains 73 sections, 109 theorems, 360 equations.

Table of Contents

  1. Introduction
  2. Key points in the proof of Theorem \ref{['thm:introrelativemmp']}
  3. Finite generation of relative adjoint rings
  4. Some aspects of the proofs of Theorems \ref{['thm:introrelativemmp']} and \ref{['thm:introfinitegen']}
  5. Preliminaries for schemes and algebraic spaces
  6. Quasi-excellence, excellence, and dualizing complexes
  7. Quasi-excellence and excellence
  8. Dualizing complexes
  9. Divisors and linear systems
  10. Divisors
  11. Linear systems
  12. Positivity, the theorem of the base, cones, and Kleiman's criterion for ampleness
  13. Relative positivity conditions
  14. Theorem of the base
  15. Cones and Kleiman's criterion for ampleness
  16. ...and 58 more sections

Key Result

Theorem A

Let $\pi\colon X \to Z$ be a locally projective morphism in one of the following categories, where $X$ and $Z$ are integral and $X$ is normal: Let $K_X$ be a canonical divisor on $X$ chosen compatibly with a dualizing complex on $Z$.For example, when $Z$ is a variety over $k$ or in cases $(setup:introcomplexanalyticgerms)$, $(setup:introberkovichspaces)$, $(setup:introrigidanalyticspaces)$, and $

Theorems & Definitions (306)

  • Theorem A: The relative minimal model program with scaling in equal characteristic zero
  • Theorem B: Finite generation of relative adjoint rings
  • Definition 2.1: EGAIV2 Définition 7.8.2 and (7.8.5); Mat80 (34.A) Definition
  • Definition 2.2: see CT20 §2.1
  • Definition 2.3: Har66 Chapter V, Definition on p. 258; Con00 p. 118; stacks-project https://stacks.math.columbia.edu/tag/0A87
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6: AB10 Definition 2.16; stacks-project https://stacks.math.columbia.edu/tag/0E4Z
  • Lemma 2.7: cf. Har66 (2) on p. 299; AB10 Proposition 2.18 and Remark on p. 14; stacks-project https://stacks.math.columbia.edu/tag/0AA3
  • proof
  • ...and 296 more