On the Morrison-Kawamata dream space and its applications

Technical Summary: On the Morrison-Kawamata Dream Space and Its Applications

Problem Statement
The paper addresses the need for a unified axiomatic framework to characterize algebraic varieties that satisfy the Morrison-Kawamata (MK) cone conjecture. While Mori dream spaces (MDS) provide a robust theory for varieties of Fano type, and Calabi-Yau (CY) type varieties are central to the study of the MK cone conjecture, there exists a gap for varieties that are neither of Fano type nor strictly Calabi-Yau type. The authors aim to develop a theory that axiomatizes the properties of varieties satisfying the MK cone conjecture, thereby extending the structural results of MDS (such as chamber decompositions and boundedness) to a broader class of varieties, including CY type varieties and others with non-trivial canonical divisors.

Methodology
The authors introduce the concept of Morrison-Kawamata Dream (MKD) spaces and MKD fiber spaces. The methodology relies on:

1. Axiomatic Definition: Defining MKD spaces via four conditions: $Q$-factoriality, the existence of good minimal models for all effective $\mathbb{R}$-Cartier divisors, the existence of a rational polyhedral cone $\Pi$ whose orbit under the pseudo-automorphism group covers the movable cone, and the "local factoriality of canonical models" for the effective cone.
2. Combinatorial Geometry: Utilizing the geometry of convex cones in Néron-Severi spaces, specifically the structure of Shokurov polytopes and the decomposition of cones into chambers where minimal models remain constant.
3. Minimal Model Program (MMP): Establishing the termination of MMPs with scaling for effective divisors within the category of MKD fiber spaces.
4. Deformation and Specialization: Employing spreading-out and specialization techniques to relate the geometry of the geometric generic fiber to the fibers over a Zariski dense subset of the base, utilizing cohomological vanishing conditions ($H^1 = H^2 = 0$) to control the behavior of cones.
5. Group Actions: Analyzing the action of pseudo-automorphism groups ($\text{PsAut}$) and automorphism groups ($\text{Aut}$) on movable, nef, and effective cones, and extending these to generic automorphism groups ($\text{GAut}$) acting on generic nef cones.

Key Contributions and Results

• Definition and Equivalence: The paper defines MKD fiber spaces and proves (Theorem 1.3) that in the absolute setting, this notion is equivalent to a set of conditions resembling MDS but adapted for birational actions (specifically, the existence of a fundamental domain for the movable cone under pseudo-automorphisms). This unifies MDS and CY type varieties (under the cone conjecture and good minimal model conjecture) into a single framework.
• Chamber Structures:
  • Shokurov Polytopes: The authors establish that for any cone generated by effective divisors where good minimal models exist, the cone decomposes into finitely many rational polyhedral chambers where the weak minimal model is constant (Theorem 1.4).
  • Nef and Effective Cones: They prove that the nef cone of effective divisors ($\text{Nef}_e$) is a rational polyhedral cone and admits a rational polyhedral fundamental domain under the action of the automorphism group (Theorem 1.5, Theorem 1.6).
• Stability under MMP and Contractions:
  • The class of MKD fiber spaces is shown to be closed under small $Q$-factorial modifications and birational contractions (Theorem 1.7).
  • A $D$-MMP with scaling of an ample divisor is proven to terminate with a good minimal model for any effective divisor $D$ on an MKD fiber space (Theorem 1.8).
• Deformation Invariance:
  • The paper establishes the generic deformation invariance of the nef, effective, and movable cones, as well as the Mori chamber decomposition, for MKD fiber spaces (Theorem 1.10). This is achieved by stratifying the base and utilizing the properties of the generic nef cone.
  • The results show that the geometric generic fiber being an MKD space implies that fibers over a Zariski open subset are "almost" MKD spaces (Theorem 6.17), satisfying the necessary cone conjectures and local factoriality properties.
• Boundedness of Moduli:
  • Applying the deformation invariance results, the authors prove a boundedness theorem (Theorem 1.11). Assuming the MK cone conjecture and the good minimal model conjecture for rationally connected CY varieties, they show that such varieties form a bounded family. This is achieved by constructing a fibration where every such variety appears as a fiber (or a birational contraction thereof).

Significance and Claims
The paper claims that MKD spaces serve as a natural generalization of Calabi-Yau type varieties in birational geometry, analogous to the role Mori dream spaces play for Fano type varieties. The significance lies in:

1. Unification: Providing a conceptual framework that extends the "gluing" of local MDS theories to varieties with trivial canonical divisors and beyond.
2. Conceptual Proofs: Demonstrating that proofs regarding cone structures and boundedness become more conceptual when viewed through the lens of MKD spaces, rather than relying on specific vanishing theorems that may not hold in the general setting.
3. Boundedness: Solving the boundedness problem for moduli spaces of rationally connected Calabi-Yau varieties under the assumption of the MK cone conjecture, a result that relies crucially on the deformation invariance of cones established for MKD fiber spaces.

The authors explicitly note that while the theory is robust, certain pathological phenomena (such as the lack of a boundedness of lengths of extremal rays in the general case) necessitate specific assumptions like the local factoriality of canonical models. The paper does not claim to resolve the MK cone conjecture itself but rather axiomatizes the consequences of its validity to facilitate further study of moduli and deformation theory.