Dual complexes of qdlt Fano type models and strong complete regularity

This paper introduces the numerical invariants of birational and strong complete regularity for Fano type pairs using qdlt models and dual complexes, establishes their fundamental properties and connections to K-stability, and proves that maximal birational strong complete regularity implies 1-complementarity while the associated jumping thresholds satisfy the ascending chain condition.

The Gist

Imagine you are an architect trying to understand the structure of a building. In the world of algebraic geometry, these "buildings" are complex shapes called varieties, and the "blueprints" are equations. Sometimes, these buildings have cracks, corners, or weird junctions called singularities.

For a long time, mathematicians had a ruler to measure how "bad" or "complicated" these cracks were. They called this measurement Regularity. If a crack was simple, the number was low; if it was complex, the number was high.

However, the authors of this paper, Jihao Liu and Konstantin Loginov, realized that this old ruler was a bit too blunt. It was like using a single ruler to measure both a tiny scratch on a watch and a collapsed bridge. It could tell you they were different, but it couldn't tell you why or give you enough detail to fix them properly.

Here is a breakdown of their new ideas using simple analogies:

1. The Problem: The "One-Size-Fits-All" Ruler

The old ruler (called Complete Regularity) was good at grouping things, but it missed the details.

• The Analogy: Imagine you have two types of broken pottery. One is a simple crack (Type A), and the other is a jagged, twisted shatter (Type D). The old ruler said, "Both of these are 'Level 1' broken."
• The Issue: In reality, Type A is easy to fix (it's "1-complementary," meaning you can patch it with one piece of clay). Type D is harder; it needs a more complex fix. The old ruler couldn't tell the difference, making it hard for architects to know which repair kit to use.

2. The Solution: A New, Sharper Ruler

The authors invented two new, super-sensitive rulers called Strong Complete Regularity and Birational Strong Complete Regularity.

• The Analogy: Instead of just measuring the size of the crack, these new rulers look at the blueprint of the building's skeleton (called the Dual Complex).
• How it works: Imagine the building is made of Lego blocks. The "Dual Complex" is a map showing how the blocks are connected.
  • If the blocks are connected in a simple line, the map is simple.
  • If they are connected in a tangled web, the map is complex.
• The Innovation: The new rulers don't just look at the current building; they look at all possible ways you could rebuild the building (using "models") to make it as clean as possible. They measure the complexity of the best possible blueprint.

3. Why "Fano Type" Matters

The paper focuses on a specific type of building called Fano type.

• The Analogy: Think of Fano types as buildings that naturally want to "pull inward" (they are positively curved, like a sphere). These are the most stable and interesting buildings in the mathematical universe.
• The Benefit: Because these buildings are so stable, the authors' new rulers work perfectly on them. They can now distinguish between the "simple crack" (Type A) and the "twisted shatter" (Type D) with perfect clarity.
  • Result: The new ruler says Type A is "Level 1" (easy to fix) and Type D is "Level 0" (harder, needs a special approach).

4. The Two Big Discoveries

The paper proves two major things about these new rulers:

A. The "One-Patch" Guarantee

• The Finding: If a building has the maximum possible score on this new ruler (meaning its skeleton is as simple as it can be), the authors prove you can always fix it with just one patch (a "1-complement").
• Why it matters: Previously, mathematicians knew that the best buildings might need one or two patches. This new result says, "No, if the blueprint is this clean, you only ever need one patch." It simplifies the repair manual significantly.

B. The "No Infinite Jumps" Rule (ACC)

• The Finding: The authors studied what happens when you slowly change the building's materials (adding a little bit of "divisor" $D$). They looked at the exact moment the ruler's number "jumps" down (meaning the building gets more complex).
• The Analogy: Imagine you are slowly adding weight to a bridge. You want to know exactly when it will start to sag. The authors proved that these "sagging points" cannot be random or infinitely close together. They follow a strict, predictable pattern.
• Why it matters: This is a huge deal for stability. It means the behavior of these mathematical buildings is orderly and predictable, not chaotic. It prevents "infinite complexity" from sneaking in.

Summary

In short, Liu and Loginov took a blunt measuring tool used for decades and replaced it with a high-definition microscope.

• They focused on the most stable, beautiful shapes in math (Fano types).
• They proved that if a shape is "perfectly simple" according to their new microscope, it is guaranteed to be easy to fix (1-complementary).
• They showed that the transition from "simple" to "complex" happens in a neat, predictable way, never in a chaotic mess.

This work helps mathematicians classify these shapes more accurately and gives them better tools to understand the fundamental structure of the universe's geometry.

Technical Summary

Here is a detailed technical summary of the paper "Dual Complexes of Qdlt Fano Type Models and Strong Complete Regularity" by Jihao Liu and Konstantin Loginov.

1. Problem Statement and Motivation

The paper addresses limitations in the existing theory of complete regularity (denoted as $Reg$) and coregularity ($Coreg$) for log canonical (lc) pairs and Fano type varieties. Introduced by Shokurov, these invariants measure the complexity of the dual complex associated with the boundary of a pair. While useful, the authors argue that complete regularity is often too coarse to distinguish between geometrically distinct singularities or varieties, particularly within the context of Fano type models and K-stability.

Key Limitations Identified:

• Indistinguishability of Types: Complete regularity fails to distinguish between different types of singularities that have different geometric structures. For example, both A-type and D-type surface singularities have complete regularity 1, despite A-type being toric and D-type not.
• Weakness for Fano Varieties: For many Fano varieties (e.g., smooth del Pezzo surfaces and most Fano threefolds), the complete regularity is maximal. This makes it ineffective for distinguishing between different families of Fano varieties or characterizing toric structures without resorting to finite covers (e.g., 2-to-1 covers to reduce D-type to A-type).
• Complementarity Gap: Pairs with maximal complete regularity are known to be either 1-complementary or 2-complementary. However, the authors seek a refined invariant that guarantees 1-complementarity for maximal cases, which is crucial for characterizing toric structures and understanding K-stability.

2. Methodology and Definitions

The authors introduce two new numerical invariants: Birational Strong Complete Regularity ($SReg_{bir}$) and Strong Complete Regularity ($SReg$). These are defined as refinements of Shokurov's complete regularity, specifically tailored to Fano type pairs.

Core Definitions:

• QF (Birational) Models: The definitions rely on the existence of qdlt Fano type models (QF models). A pair $(Y/Z, B_Y)$ is a QF model of $(X/Z, B)$ if:
  1. There is a birational map $g: Y \dashrightarrow X$ over $Z$.
  2. $(Y, B_Y)$ is qdlt (quasi-divisorial log terminal).
  3. $-(K_Y + B_Y)$ is ample over $Z$.
  4. Technical Condition: Any $g$-exceptional prime divisor $D$ with multiplicity 0 in $B_Y$ does not contain any lc center of $(Y, B_Y)$. This condition ensures that the dual complex captures the relevant lc geometry without "inessential" exceptional divisors interfering.
• The Invariants:
  • $SReg_{bir}(X/Z, B) := \max \{ -1, \dim \mathcal{D}(Y, B_Y) \}$, where the maximum is taken over all QF birational models.
  • $SReg(X/Z, B) := \max \{ -1, \dim \mathcal{D}(Y, B_Y) \}$, where the maximum is taken over all QF morphism models.
  • Here, $\mathcal{D}(Y, B_Y)$ denotes the dual complex of the reduced boundary.
• Strong Coregularity: Defined as $SCoreg = \dim X - 1 - SReg$.

Technical Framework:

The authors develop a robust machinery to work with these invariants:

• Reduced QF Models: They introduce "reduced" models where the boundary is canonical ($B_Y \ge g^{-1}_* B + \text{Exc}(g)$), allowing for better control over the dual complex.
• QAG and QAA Models: They define Qdlt Anti-Good (QAG) and Qdlt Anti-Ample (QAA) models. These serve as generalizations of Kollár models (used in local K-stability) to the global and relative settings.
• Correspondence Theorems: They establish a correspondence showing that $SReg_{bir}$ can be computed using QAG or QAA models, and that these models preserve the PL-homeomorphism class of the dual complex.

3. Key Contributions and Results

The paper establishes two main theorems that highlight the superior behavior of the new invariants compared to classical complete regularity.

Result 1: Maximal Strong Complete Regularity Implies 1-Complementarity

Theorem 1.5 (Theorem 4.1):
Let $(X/Z \ni z, B)$ be a pair. If the birational strong complete regularity is maximal, i.e.,
$$SReg_{bir}(X/Z \ni z, B) = \dim X - 1 - \dim z$$
Then:

1. The pair is 1-complementary.
2. There exists a 1-complement $(X/Z \ni z, B^+)$ such that the regularity of the complement is also maximal.

Significance: Unlike classical complete regularity (where maximal cases can be 1- or 2-complementary), maximal strong complete regularity always implies 1-complementarity. This provides a precise characterization for toric singularities and Fano varieties without needing finite covers. The proof utilizes induction on dimension, inversion of adjunction, and the construction of monotonic 1-complements on reduced QF models.

Result 2: ACC for Strong Complete Regularity Thresholds

Theorem 1.7:
Let $\Gamma \subset [0, +\infty)$ be a set satisfying the Descending Chain Condition (DCC). The set of strong complete regularity thresholds satisfies the Ascending Chain Condition (ACC).
Specifically, for a pair $(X/Z, B)$ and a divisor $D$, consider the function $SReg(t) = SReg(X/Z, B + tD)$. The values of $t$ where $SReg(t)$ drops below an integer $n$ form a set that satisfies ACC.

Significance: This extends the famous ACC for lc thresholds (HMX) and ACC for R-complementary thresholds to the context of strong complete regularity. It demonstrates that the "jumping points" of these invariants are well-behaved and discrete, a crucial property for boundedness results in the Minimal Model Program (MMP).

4. Significance and Impact

• Refinement of Invariants: The paper successfully refines the coarse invariant of complete regularity into a tool sensitive enough to distinguish between A-type and D-type singularities (A-type has $SReg=1$, D-type has $SReg=0$).
• Connection to K-Stability: The definitions are motivated by and aligned with K-stability theory. The "QF models" and the technical condition (4) are designed to mirror the behavior of Kollár valuations and special valuations in local K-stability. The results suggest that the dual complexes of these models carry richer intrinsic structures relevant to stability.
• Boundedness and Classification: The ACC result for thresholds is a foundational step toward proving boundedness results for Fano type varieties with specific regularity constraints. It allows for the classification of singularities and pairs based on their "strong" regularity properties.
• Toric Characterization: The results provide a pathway to characterize toric varieties and singularities via 1-complementarity and dual complex dimensions without the need for covering tricks, which was a limitation in previous approaches.

5. Conclusion

Liu and Loginov introduce Strong Complete Regularity as a powerful new invariant for Fano type pairs. By leveraging qdlt Fano type models and dual complexes, they prove that maximal strong regularity guarantees 1-complementarity and that the thresholds of these invariants satisfy the ACC. These results bridge the gap between the classification of singularities, the theory of complements, and the geometric structures underlying K-stability, offering a more precise tool for the study of Fano varieties and log Calabi-Yau pairs.