Families of stable 3-folds in positive characteristic

The paper demonstrates that flat families of stable 3-folds fail to form proper moduli spaces in any positive characteristic $p>0$, a result that simultaneously yields log canonical 4-fold pairs with non-weakly normal log canonical centers.

The Gist

Imagine you are an architect trying to build a perfect, infinite library of all possible "stable" 3-dimensional shapes (called 3-folds) in mathematics. You want to organize them into a neat catalog (a moduli space) where every shape has a specific spot, and if you slowly morph one shape into another, the catalog stays complete and orderly.

In the world of mathematics with characteristic 0 (think of this as our standard, "normal" reality), this library works beautifully. You can take a smooth shape, let it degenerate (crumble slightly) into a singular shape, and the catalog knows exactly where to put it. The rules are consistent.

However, János Kollár's paper reveals a shocking discovery: In "positive characteristic" (a mathematical universe that behaves like arithmetic modulo a prime number $p$), this library collapses.

Here is the breakdown of the paper using simple analogies:

1. The Broken Blueprint (The Main Problem)

In our standard world, if you have a family of shapes changing over time, you can predict their final, "canonical" form (the most stable version) using a specific set of rules. It's like having a blueprint that says, "If you start with this smooth house and let the foundation settle, it will always become this specific type of cottage."

Kollár shows that in positive characteristic, this blueprint fails for 3D shapes.

• The Glitch: You can start with a family of perfect, smooth 3D shapes. As you move toward a specific point in time (the "special fiber"), the shapes change.
• The Surprise: When you try to apply the standard rules to find the "final stable version" of that specific point, the result is a mathematical disaster. The resulting shape doesn't fit the rules of the library. It's not "stable" in the way the catalog requires.
• The Analogy: Imagine you are baking a cake. In normal physics, if you follow the recipe, you get a cake. In this strange mathematical universe, if you follow the exact same recipe for a 3D cake, the oven suddenly produces a shape that is so broken it can't even be recognized as a cake anymore. The "moduli space" (the catalog) has a hole in it.

2. The "Jumping" Numbers (The Plurigenera)

How did he prove this? He looked at something called plurigenera.

• The Concept: Think of plurigenera as a "complexity score" or a "count of patterns" on the shape. For a stable family, this score should be constant or change very smoothly.
• The Anomaly: Kollár constructed a family where, for almost all moments in time, the complexity score is low. But at one specific moment (the special fiber), the score jumps dramatically.
• The Metaphor: Imagine a river flowing smoothly. The water level (complexity) is steady. But at one specific rock, the water suddenly spikes up to a massive height, then drops back down. In standard math, this "spike" shouldn't happen in a stable family. Kollár found a way to force this spike using the unique properties of positive characteristic.

3. The "Unipotent" Trick (The Secret Ingredient)

How did he make the complexity jump? He used a mathematical object called a unipotent vector bundle on an elliptic curve (a doughnut-shaped surface).

• The Analogy: Imagine a rubber band (the bundle) wrapped around a doughnut. In normal math, if you stretch it, it behaves predictably. But in positive characteristic, there is a special "glitch" where you can stretch the rubber band in a way that makes it suddenly snap into a different, more complex configuration only at a specific point.
• Kollár used this "glitchy" rubber band to build a 2D surface, and then stacked it to build a 3D shape. This allowed the "complexity score" to jump unexpectedly.

4. The "Ghost" Shapes (Non-Weakly Normal Centers)

One of the side effects of this discovery is the creation of strange 4-dimensional shapes (4-folds).

• The Issue: In these shapes, there are "centers" (specific points or lines) that are supposed to be well-behaved. But in this new world, they are "not weakly normal."
• The Metaphor: Imagine a building with a central pillar. In a normal building, the pillar is solid and continuous. In Kollár's example, the pillar exists, but it has a "ghost" quality—it's there, but it's not fully connected in the way it should be. It's like a pillar made of smoke that looks solid from a distance but falls apart if you try to touch it. This breaks the rules of how we usually classify these shapes.

5. Why Does This Matter?

You might ask, "Who cares about math in positive characteristic?"

• The Reality: This isn't just abstract theory. Positive characteristic is crucial for number theory and cryptography.
• The Lesson: For a long time, mathematicians hoped that the "weirdness" of positive characteristic was just a technical nuisance that could be fixed with better tools. They thought, "If we just account for the weirdness, the rules will be the same as in normal math."
• The Verdict: Kollár's paper says no. The rules are fundamentally different. The "library" of stable 3D shapes simply cannot be built in the same way in this universe. We cannot just patch the holes; the entire foundation needs to be reconsidered.

Summary

János Kollár built a mathematical "trap." He showed that if you try to organize 3D shapes in a world with modular arithmetic (positive characteristic), the standard rules of stability break down. The shapes behave unpredictably, their complexity jumps unexpectedly, and the resulting "catalog" of shapes is incomplete. It proves that the geometry of these worlds is far more subtle and dangerous than we previously thought.

Technical Summary

1. Problem Statement

The paper addresses the construction of moduli spaces for stable varieties in positive characteristic ($p > 0$).

• Context: In characteristic 0, the moduli space of varieties of general type is compactified using the Minimal Model Program (MMP). One starts with a smooth family, extends it to a semi-stable family, and then takes the relative canonical model. The fibers of this relative model are "stable varieties" (having semi-log-canonical singularities and an ample canonical class). This process yields a proper moduli space.
• The Issue: It was an open question whether this standard procedure works in positive characteristic for dimensions $\ge 3$. While the MMP for 3-folds is known to work for $p \ge 7$, the behavior of families and the existence of proper moduli spaces in arbitrary positive characteristic remained unclear.
• Specific Question: Do flat families of stable 3-folds of fixed volume form a proper moduli theory in any characteristic $p > 0$?

2. Methodology

Kollár constructs explicit counterexamples using a combination of elliptic ruled surfaces, unipotent vector bundles, and cone constructions. The methodology proceeds in three main stages:

A. Unipotent Bundles on Elliptic Curves (Characteristic $p$)

The core of the construction relies on the behavior of unipotent vector bundles on elliptic curves in positive characteristic.

• Let $E$ be an elliptic curve. There exists a unique indecomposable unipotent vector bundle $F_2$ of rank 2 fitting into a non-split extension: $0 \to \mathcal{O}_E \to F_2 \to \mathcal{O}_E \to 0$.
• Key Phenomenon: In characteristic 0, pulling back $F_2$ via a map $\tau: E' \to E$ preserves non-splitness unless the map is trivial. However, in characteristic $p$, there exist degree $p$ maps $\rho: E' \to E$ such that the induced map on cohomology $\rho^*: H^1(E, \mathcal{O}_E) \to H^1(E', \mathcal{O}_{E'})$ is zero.
• Consequently, the pullback $\rho^* F_2$ splits into $\mathcal{O}_{E'} \oplus \mathcal{O}_{E'}$. This splitting causes a "jump" in the number of global sections ($h^0$) of certain divisors associated with these bundles.

B. Construction of a Pathological Family of Surfaces (Example 8)

Using the above phenomenon, Kollár constructs a family of elliptic surfaces $g: S \to \mathbb{A}^1$:

• The generic fiber $S_t$ ($t \neq 0$) is associated with the non-split bundle $F_2$.
• The special fiber $S_0$ is associated with the split bundle $\mathcal{O} \oplus \mathcal{O}$.
• By constructing a specific divisor $D$ and a pair $(S, \Delta + D)$, he ensures that the plurigenera (dimensions of spaces of pluricanonical forms) jump at $t=0$. Specifically, $h^0(S_0, m(K_{S_0} + \dots)) > h^0(S_t, m(K_{S_t} + \dots))$ for large $m$.
• This jump violates the deformation invariance of plurigenera expected in characteristic 0.

C. Cone Construction to Dimension 3 (Example 4)

To lift this pathology to 3-folds, Kollár employs a cone construction (inspired by McKernan):

1. Start with the surface pair $(S, \Delta + D)$ from Step B.
2. Form the projective cone $C$ over $S$ with vertex $v$.
3. Blow up the vertex to obtain a $\mathbb{P}^1$-bundle $Y \to S$.
4. Construct a pair $(Y, \Delta_Y + Z_0 + Z'_\infty + D_Y)$ where $Z_0$ corresponds to the base $S$.
5. Crucial Step: The behavior of the pluricanonical maps of the base $S$ is "visible" in the pluricanonical maps of the 3-fold $Y$. Because the plurigenera of $S$ jump at $t=0$, the plurigenera of the 3-fold family $Y \to \mathbb{A}^1$ also jump.

3. Key Contributions and Results

Theorem 1: Failure of Proper Moduli

The main result is that flat families of stable 3-fold pairs of fixed volume do not form a proper moduli theory in any characteristic $p > 0$.

• Mechanism of Failure: The author constructs a family where the generic fibers are smooth with ample canonical class. The relative canonical model exists, but its special fiber (the limit) fails Serre's condition $S_2$.
• Normalization Issue: While the normalization of the special fiber is a stable variety, the family itself does not behave well under the standard moduli compactification process. The "canonical model" of the family is not flat over the base.

Corollary 5: Non-Weakly Normal Log Canonical Centers

As a byproduct, the paper constructs log canonical 4-fold pairs $(X, S + \Delta)$ where the log canonical center $S$ is a Cartier divisor that is not weakly normal.

• In characteristic 0, all log canonical centers are seminormal (and thus weakly normal).
• This result shows that new, subtle pathologies appear in positive characteristic even for dimensions where the MMP is otherwise well-behaved (specifically for 4-folds).

Lemma 17: Non-Cohen-Macaulay (Non-CM) Singularities

The paper demonstrates that the special fibers of the constructed families are not Cohen-Macaulay (CM).

• In characteristic 0, being CM is a deformation-invariant property for stable varieties.
• In the constructed examples, the generic fibers are CM, but the special fiber is not. This non-CM property is the root cause of the failure of the moduli space to be proper.

4. Significance and Implications

1. Refutation of Naive Expectations: It was hoped that once the MMP and resolution of singularities were established in positive characteristic, the moduli theory would mirror the characteristic 0 case. This paper proves that new, fundamental obstructions exist in positive characteristic that are not merely technical hurdles.
2. Dimensional Threshold: The examples show that these pathologies arise in dimension 3 (and 4 for the corollary). The dimension of the examples is roughly twice the characteristic ($2p$), suggesting a deep link between the failure of Kodaira vanishing and the geometry of unipotent bundles.
3. Moduli Theory Limitations: The result implies that the standard definition of "stable variety" (requiring $S_2$ and semi-log-canonical singularities) is insufficient for a proper moduli space in positive characteristic. One might need to allow families where fibers are only stable after normalization, but a precise conjecture for such a theory is currently unknown.
4. Weak Normality: The discovery of non-weakly normal log canonical centers challenges the assumption that properties holding in characteristic 0 (like the normality of centers) automatically extend to positive characteristic, even in high dimensions.

Conclusion

János Kollár demonstrates that the compactification of the moduli space of stable 3-folds fails in positive characteristic due to the existence of families where the relative canonical model is not flat. This failure stems from the non-deformation-invariance of the Cohen-Macaulay property, driven by the specific behavior of unipotent vector bundles on elliptic curves in characteristic $p$. The paper establishes that the moduli theory in positive characteristic is significantly more subtle than in characteristic 0, requiring new definitions or constraints that are not yet fully understood.