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On the Morrison-Kawamata dream space and its applications

Sung Rak Choi, Xingying Li, Zhan Li, Chuyu Zhou

TL;DR

The paper defines Morrison-Kawamata dream spaces (MKD spaces) as a broad birational framework that generalizes Mori dream spaces while accommodating Calabi–Yau type phenomena under the Morrison–Kawamata cone conjecture. It develops foundational tools—Shokurov polytopes, polyhedral fundamental domains, and MMP with scaling—showing that birational contractions preserve MKD structure and that key cones deform predictably in families. It then demonstrates deformation-invariance results for nef, movable, and effective cones, as well as Mori chamber decompositions, and applies these to boundedness questions for moduli spaces and complements. Overall, MKD spaces extend Mori dream-space techniques to a wider class of varieties, enabling robust control of birational models, cone structures, and moduli in families.

Abstract

We develop the theory of Morrison-Kawamata dream spaces, which axiomatizes varieties (not necessarily of Calabi-Yau type) that satisfy the Morrison-Kawamata cone conjecture. Using this theory, we establish the generic deformation invariance of various cones and apply it to the boundedness problem of algebraic varieties.

On the Morrison-Kawamata dream space and its applications

TL;DR

The paper defines Morrison-Kawamata dream spaces (MKD spaces) as a broad birational framework that generalizes Mori dream spaces while accommodating Calabi–Yau type phenomena under the Morrison–Kawamata cone conjecture. It develops foundational tools—Shokurov polytopes, polyhedral fundamental domains, and MMP with scaling—showing that birational contractions preserve MKD structure and that key cones deform predictably in families. It then demonstrates deformation-invariance results for nef, movable, and effective cones, as well as Mori chamber decompositions, and applies these to boundedness questions for moduli spaces and complements. Overall, MKD spaces extend Mori dream-space techniques to a wider class of varieties, enabling robust control of birational models, cone structures, and moduli in families.

Abstract

We develop the theory of Morrison-Kawamata dream spaces, which axiomatizes varieties (not necessarily of Calabi-Yau type) that satisfy the Morrison-Kawamata cone conjecture. Using this theory, we establish the generic deformation invariance of various cones and apply it to the boundedness problem of algebraic varieties.

Paper Structure

This paper contains 30 sections, 52 theorems, 310 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and terminology
  4. Minimal models of $\mathbb{R}$-Cariter divisors
  5. Cones in Néron-Severi spaces and the Morrison-Kawamata cone conjecture
  6. Geometry of convex cones
  7. Foundations on Morrison-Kawamata dream spaces
  8. Definition of Morrison-Kawamata dream spaces
  9. Discussion on local factoriality of canonical models
  10. Shokurov polytopes of minimal models
  11. Fundamental domains for nef cones and effective cones
  12. MKD spaces under birational contractions
  13. Minimal model program for MKD spaces
  14. Extremal contractions and flips in the category of MKD spaces
  15. MMP with scaling for MKD spaces
  16. ...and 15 more sections

Key Result

Theorem 1.2

Let $X$ be a normal projective variety. Assume that an effective $\mathbb{R}$-Cartier divisor admits a minimal model. Then $X$ is an MKD space if and only if it satisfies the following conditions:

Figures (1)

  • Figure 1: Neighborhood of $D$ in the context of the local factoriality of canonical models

Theorems & Definitions (118)

  • Definition 1.1: Mori dream space HK00
  • Definition 1.1: Morrison-Kawamata dream fiber space
  • Theorem 1.2
  • Theorem 1.3: Shokurov polytope for minimal models
  • Theorem 1.4: Shokurov polytope for nef cones
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 108 more