MMP for Enriques pairs and singular Enriques varieties

This paper introduces the class of primitive Enriques varieties, demonstrates their stability under the Minimal Model Program by proving that $(K_Y+B_Y)$-MMPs for Enriques manifolds terminate with minimal models having canonical singularities, and investigates their asymptotic theory.

The Gist

Imagine you are an architect trying to understand the fundamental building blocks of a very strange, high-dimensional universe. In the world of algebraic geometry, these "universes" are shapes called varieties. Some are smooth and perfect, like a polished marble sphere; others are crumpled, folded, or have sharp corners.

This paper is a guidebook for navigating these shapes, specifically focusing on a special family called Enriques varieties. Think of them as the "cousins" of a famous family called K3 surfaces and Enriques surfaces (which are 2D shapes), but living in much higher dimensions (4D, 6D, and beyond).

Here is the story of the paper, broken down into simple concepts with some creative metaphors.

1. The Characters: The "Smooth" vs. The "Rough"

• The IHS Manifold (The Perfect Twin): Imagine a perfectly smooth, simply connected shape (like a sphere) that has a special "symplectic" property. In math-speak, this is an Irreducible Holomorphic Symplectic (IHS) manifold. It's the "gold standard" of smoothness.
• The Enriques Manifold (The Quotient): Now, imagine you take that perfect IHS shape and fold it over itself a few times, gluing points together in a specific way. The result is an Enriques manifold. It's not simply connected (it has "holes" or loops you can't shrink), but it's still smooth. It's like taking a perfect sheet of paper and rolling it into a cylinder; the paper is the IHS, the cylinder is the Enriques.
• The Problem: Real life (and math) isn't always smooth. Sometimes these shapes get crumpled or develop singularities (sharp points). The authors wanted to know: What happens if we allow these Enriques shapes to be rough or singular?

2. The Mission: The "Renovation" (The MMP)

The paper tackles a massive project called the Minimal Model Program (MMP).

• The Metaphor: Imagine you have a messy, cluttered house (a complex geometric shape). You want to renovate it to be as simple and efficient as possible without tearing down the whole structure. You want to remove unnecessary walls, fix cracks, and get rid of "bad" corners, but you must keep the house standing.
• The Process: In math, this involves a series of steps:
  1. Flips: Swapping a small part of the shape for a slightly different configuration to make it "nicer."
  2. Contractions: Collapsing a part of the shape into a point or a line.
• The Big Question: Does this renovation process ever stop? Or does it go on forever, flipping and contracting endlessly?
  • For smooth shapes, we knew the answer.
  • For these rough, singular Enriques shapes, nobody knew for sure.

3. The Breakthrough: "Primitive Enriques Varieties"

The authors introduced a new definition: Primitive Enriques Varieties.

• The Analogy: Think of the smooth Enriques manifold as a pristine, high-end hotel. The "Primitive Enriques Variety" is the same hotel after it has been through a storm, has some broken windows, and maybe a few cracked walls, but it still retains the essential soul of the hotel.
• The Discovery: They proved that no matter how messy you start with (as long as it's a specific type of "log canonical" mess), if you run the renovation process (MMP), it will always stop.
• The Result: You will end up with a "Minimal Model." This final shape is:
  • Q-factorial: It's "fixable" (mathematically well-behaved).
  • Primitive Enriques: It's still part of the same family, just the "best" version of the rough version.
  • Canonical: It has the best possible kind of singularities (the "least bad" kind of cracks).

Why is this huge? It confirms a major conjecture in math: that for this entire class of shapes, the renovation process always terminates. You never get stuck in an infinite loop of fixes.

4. The Strategy: The "Mirror Trick"

How did they prove it? They used a clever trick involving coverings.

• The Metaphor: Imagine you are trying to fix a broken Enriques shape (the cylinder). It's hard to see the cracks clearly. But you know this cylinder is just a folded version of a perfect sphere (the IHS manifold).
• The Move: Instead of trying to fix the cylinder directly, they "unfolded" it back into the perfect sphere.
  1. They lifted the renovation steps from the rough Enriques shape up to the smooth IHS sphere.
  2. They proved that the renovation on the sphere definitely stops (because we already knew that for smooth shapes).
  3. They then "folded" the result back down.
• The Logic: If the renovation stops on the perfect sphere, it must stop on the rough cylinder, because the cylinder is just a shadow of the sphere. This "lifting" technique was the key to solving the puzzle.

5. The Bonus: Measuring the "Volume"

The second half of the paper looks at the asymptotic theory.

• The Metaphor: Imagine you have a garden (the shape) and you want to plant flowers (divisors). You want to know: "If I plant more and more flowers, how fast does the garden fill up?"
• The Finding: They calculated the "volume" of these shapes. They found that the volume function behaves like a piecewise polynomial.
  • Translation: Imagine a landscape that is flat in some areas, slopes up in others, and curves down in others. The "volume" of the shape changes in a predictable, mathematical way depending on which "zone" of the landscape you are in. It's not random chaos; it follows a strict, beautiful pattern.

Summary

In plain English, this paper says:

"We have defined a new class of rough, high-dimensional shapes called Primitive Enriques Varieties. We proved that if you try to simplify them using standard mathematical tools, the process always finishes, leaving you with a clean, well-behaved version of the shape. We did this by using a 'mirror trick' to look at their smooth, perfect twins. Finally, we figured out exactly how the 'size' of these shapes grows, showing it follows a predictable, patterned rule."

It's a story of taking something messy and proving that, deep down, it follows a perfect, orderly plan.

Technical Summary

Here is a detailed technical summary of the paper "MMP for Enriques Pairs and Singular Enriques Varieties" by Francesco Antonio Denisi, Ángel David Ríos Ortiz, Nikolaos Tsakanikas, and Zhixin Xie.

1. Problem Statement and Context

The paper addresses two primary objectives within the framework of the Minimal Model Program (MMP) and the classification of higher-dimensional algebraic varieties:

1. Termination of MMP for Enriques Pairs: While the termination of flips is a central conjecture in the MMP, it is fully resolved only in dimension 3 and partially in dimension 4. For specific classes of varieties, such as Irreducible Holomorphic Symplectic (IHS) manifolds, termination results are known (e.g., Lehn and Pacienza, 2016). The authors aim to extend these results to Enriques manifolds (projective manifolds that are not simply connected but have an IHS universal cover) and their singular analogs. Specifically, they seek to prove that any $(K_Y + B_Y)$-MMP for an Enriques pair $(Y, B_Y)$ terminates with a minimal model.
2. Characterization of Singular Models: The authors aim to characterize the underlying variety of the resulting minimal model. They introduce and study Primitive Enriques Varieties as the singular analogs of Enriques manifolds, analogous to how Primitive Symplectic varieties generalize IHS manifolds.
3. Asymptotic Theory: The paper investigates the asymptotic properties of divisors on Enriques manifolds, specifically the volume function and the duality between cones of $k$-ample divisors and $k$-movable curves.

2. Methodology

The authors employ a sophisticated strategy involving equivariant MMP and Galois quasi-étale covers to lift problems from Enriques varieties to their IHS covers.

• Lifting to the Universal Cover: Given an Enriques manifold $Y$ with universal cover $\gamma: X \to Y$ (where $X$ is an IHS manifold), the authors lift the MMP on $Y$ to an equivariant MMP on $X$.
  • They utilize the fact that the fundamental group $\pi_1(Y)$ acts on $X$.
  • Key Technical Step (Proposition 3.3): They prove that any usual MMP step on $Y$ can be lifted to a $\pi_1(Y)$-equivariant MMP step on $X$.
  • Key Technical Step (Proposition 3.4): They establish that the termination of a usual MMP implies the termination of the corresponding equivariant MMP. This allows them to deduce the termination of the Enriques MMP from the known termination results for IHS manifolds (Theorem 1.1).
• Definition of Singular Varieties: The authors define Primitive Enriques Varieties as quotients of Primitive Symplectic varieties by finite non-symplectic group actions that are free in codimension one. They distinguish these from Irreducible Enriques Varieties (quotients of Irreducible Symplectic varieties), noting that the former class is more robust under birational morphisms.
• Asymptotic Analysis: For the asymptotic theory, they utilize the Nakayama–Zariski decomposition and the Beauville–Bogomolov–Fujiki (BBF) form on the universal cover to analyze the volume function and base loci. They relate the geometry of the Enriques manifold to its IHS cover via pullback and pushforward of divisors and curves.

3. Key Contributions and Definitions

• Primitive Enriques Varieties (Definition 5.11): A normal projective variety $Y$ is a primitive Enriques variety if it is the quotient $X/G$, where $X$ is a primitive symplectic variety and $G$ is a finite non-symplectic group acting freely in codimension one.
  • Significance: This class includes Enriques manifolds as smooth members but also allows for singularities (canonical or klt). Crucially, unlike Irreducible Enriques varieties, the class of Primitive Enriques varieties is preserved under the MMP (Proposition 5.19).
• Irreducible Enriques Varieties (Definition 5.7): Defined as GGK varieties (quotients of Irreducible Symplectic varieties) where the holonomy group is strictly smaller than the restricted holonomy. These correspond to the "irreducible" case but are not preserved under all birational contractions.
• Equivariant MMP Lifting: The paper provides a rigorous framework for lifting MMP steps from a quotient $Y = X/G$ to the cover $X$, ensuring that the group action is preserved throughout the process.

4. Main Results

Theorem 1.2 (Termination of MMP for Enriques Pairs)

Let $Y$ be an Enriques manifold and $B_Y$ an effective $\mathbb{R}$-divisor such that $(Y, B_Y)$ is log canonical. Then any $(K_Y + B_Y)$-MMP terminates with a minimal model $(Y', B_{Y'})$, where:

• $Y'$ is a $\mathbb{Q}$-factorial primitive Enriques variety with canonical singularities.
• $B_{Y'}$ is a nef and effective $\mathbb{R}$-divisor.
• Significance: This confirms the termination of flips conjecture specifically for the class of Enriques manifolds and identifies the precise nature of the singular minimal models.

Theorem 1.3 (Volume Function)

If $Y$ is an Enriques manifold of dimension $2n$, its volume function$\text{vol}Y: N_1(Y)\mathbb{R} \to \mathbb{R}_{\geq 0}$ is:

• Homogeneous of degree $2n$.
• Piecewise polynomial.
• Method: The proof relies on the decomposition of the big cone into Boucksom–Zariski chambers, lifted from the IHS universal cover where the volume is known to be polynomial.

Theorem 1.4 (Duality of Cones)

For an Enriques manifold $Y$ of dimension $2n$and any integer$k \in {1, \dots, 2n}$:
$$\text{Amp}_k(Y)^\vee = \overline{\text{bMob}_k(Y)}$$
where $\text{Amp}_k(Y)$ is the cone of $k$-ample classes and $\overline{\text{bMob}_k(Y)}$ is the closure of the cone of birationally $k$-movable curves.

• Specific Case: If $1 \leq k \leq n$, then$\text{Amp}_k(Y)^\vee = \text{NE}(Y)$ (the cone of effective curves).
• Significance: This generalizes Payne's question and the duality results for IHS manifolds (DRO25) to the Enriques setting, utilizing the MMP lifting techniques to handle the singular models required for the proof.

5. Examples and Structural Insights

The paper constructs various examples to illustrate the differences between smooth Enriques manifolds and their singular counterparts:

• Simply Connected Singular Varieties: Unlike smooth Enriques manifolds (which have $\pi_1 \neq 1$), singular Primitive Enriques varieties can be simply connected (e.g., quotients of K3 surfaces with fixed points).
• Uniruledness: Singular Primitive Enriques varieties can be uniruled (e.g., if they have klt but non-canonical singularities), whereas smooth Enriques manifolds are never uniruled.
• Canonical Class: While smooth Enriques manifolds have torsion canonical class, singular Primitive Enriques varieties can have trivial canonical class ($K_Y \sim 0$) or strictly torsion canonical class, depending on the parity of the dimension and the group action.

6. Significance

• Advancement of MMP: The paper resolves the termination of flips for a significant new class of varieties (Enriques manifolds) by leveraging the structure of their universal covers.
• Singular Geometry: It establishes a robust theory for singular Enriques varieties, defining a class (Primitive Enriques) that behaves well under the MMP, filling a gap in the classification of varieties with numerically trivial canonical divisors.
• Asymptotic Geometry: The results on volume functions and cone duality extend the powerful tools of asymptotic algebraic geometry from the smooth IHS setting to Enriques manifolds, providing a complete picture of the positivity properties of divisors in this context.
• Unification: The work unifies the study of IHS manifolds, Enriques manifolds, and their singular analogs under a single framework of equivariant MMP and Galois covers.