Notes on rational chain connectedness

This paper extends Hacon–McKernan's rational chain connectedness theorem to the complex analytic setting using the minimal model program, thereby proving that fibers of resolutions of complex analytic Kawamata log terminal singularities are rationally chain connected without relying on extension theorems.

The Gist

Here is an explanation of Osamu Fujino's paper, "Notes on Rational Chain Connectedness," translated into everyday language using analogies.

The Big Picture: Untangling a Messy Knot

Imagine you have a giant, tangled ball of yarn (this represents a complex geometric shape called a variety). In mathematics, specifically in a field called algebraic geometry, researchers want to know: Is this ball of yarn actually one single, connected piece, or is it a bunch of separate strings glued together?

Even better, they want to know if you can travel from any point on the yarn to any other point by walking along a path made entirely of straight lines (or circles, in the mathematical sense). If you can do this, the shape is called "rationally connected."

If the shape has some "knots" or "bad spots" (mathematicians call these singularities), the question becomes harder. Can you still travel between points, perhaps by hopping over the bad spots? This is called "rational chain connectedness."

The Problem: The "Magic Spell" Was Too Hard

A few years ago, two mathematicians named Hacon and McKernan proved a famous theorem about this. They showed that if a shape has certain "good" properties (specifically, if it's a "Fano" shape, which is like a balloon that curves inward everywhere), then it is indeed rationally chain connected.

However, their proof relied on a very difficult, complex tool called an "Extension Theorem."

• The Analogy: Imagine trying to fix a broken vase. Hacon and McKernan's method was like using a super-advanced, magical glue that only a handful of wizards in the world knew how to mix. It worked perfectly, but it was so complicated that even other wizards found it hard to remember the recipe. If you made one tiny mistake in the mixing, the whole thing fell apart.

Fujino's Solution: The "Construction Crew"

Osamu Fujino, the author of this paper, decided to solve the same problem but without using that magical glue. Instead, he used a standard, reliable construction method known as the Minimal Model Program (MMP).

• The Analogy: Instead of using the magical glue, Fujino used a team of construction workers with standard tools (hammers, saws, and blueprints).
  • The Minimal Model Program is like a step-by-step construction guide. It says: "If your building is messy, knock down the extra walls, flip the roof if it's upside down, and keep doing this until you have a simple, stable structure."
  • Fujino showed that by following these standard construction steps, you can prove the shape is connected without needing the "magic glue."

Why is this important?

1. Accessibility: It's easier for other mathematicians to understand and use because it relies on standard tools rather than obscure, complex tricks.
2. New Territory: The original proof only worked for "algebraic" shapes (shapes defined by polynomial equations). Fujino extended this proof to complex analytic spaces.
   • The Analogy: The original proof worked for shapes built out of Lego bricks (algebraic). Fujino proved it also works for shapes made of flowing water or flexible rubber (complex analytic spaces), which are much harder to pin down.

The Main Result: What Did He Actually Prove?

Fujino proved a specific rule about resolutions of singularities.

• The Scenario: Imagine you have a crumpled piece of paper with a sharp crease (a singularity). You want to smooth it out (resolve the singularity) so it becomes a nice, flat sheet.
• The Question: When you smooth it out, does the "extra stuff" you added to fix the crease form a connected path?
• The Answer: Yes. Fujino proved that if you take a "kawamata log terminal" shape (a specific type of geometric object with mild bad spots) and smooth it out, the fibers (the layers of the smoothing process) are rationally chain connected.

In plain English: If you fix a geometric shape that has some minor defects, the "patching material" you use to fix it is always a single, connected web of paths. You can walk from any part of the patch to any other part without getting stuck.

The "Chain" in Rational Chain Connectedness

To understand the title, imagine a chain made of links.

• Rational Connectedness: You can walk from Point A to Point B on a single, unbroken straight line.
• Rational Chain Connectedness: You can walk from Point A to Point B, but you might have to hop from one straight line to another. As long as the lines are connected to each other (like a chain), you can get there.

Fujino's work guarantees that even if the shape is messy or has "bad spots," you can always find a chain of straight paths to get from anywhere to anywhere else.

Summary of the Paper's Journey

1. Introduction: Fujino says, "Hacon and McKernan proved this great thing, but their proof was too hard to follow. Let's do it again using standard tools."
2. The Toolkit: He sets up the rules of the "Minimal Model Program" (the construction crew) for complex shapes.
3. The Proof: He runs the construction crew through the shape. He shows that no matter how you try to break the shape apart, the "bad spots" and the "fixes" are always linked together in a chain.
4. The Result: He proves that the fibers of any resolution of these shapes are connected.
5. The Benefit: This makes the theory more robust and easier for the next generation of mathematicians to use, especially when dealing with shapes that aren't just simple polynomials but more complex, flowing geometric objects.

In a nutshell: Fujino took a difficult mathematical puzzle, threw away the "magic wand" that made it hard to understand, and solved it using a reliable, step-by-step construction manual. He showed that even in the messy, complex world of analytic geometry, everything is connected by a chain of paths.

Technical Summary

Here is a detailed technical summary of the paper "Notes on Rational Chain Connectedness" by Osamu Fujino.

1. Problem Statement

The paper addresses the Rational Chain Connectedness (RCC) of fibers in the context of complex analytic geometry. Specifically, it aims to generalize the main results of Hacon and McKernan's seminal work ([HM2], 2007) from the algebraic setting (projective varieties over $\mathbb{C}$) to the complex analytic setting (projective morphisms between complex analytic spaces).

The core problem involves proving that under specific positivity conditions on the anti-canonical divisor (or a log-canonical pair), the fibers of a morphism are rationally chain connected modulo the non-kawamata log terminal (non-klt) locus.

Key Context:

• Original Approach: Hacon and McKernan's original proof relied heavily on intricate extension theorems (specifically from [HM1]), which are technically demanding and difficult to apply directly in the analytic category without significant modification.
• Goal: To provide a proof in the complex analytic setting that avoids the direct use of these complex extension theorems, relying instead on the Minimal Model Program (MMP) developed for complex analytic spaces.

2. Methodology

Fujino's approach diverges from the original Hacon–McKernan strategy by shifting the technical burden from extension theorems to the machinery of the Minimal Model Program (MMP) in the complex analytic category.

• Framework: The paper utilizes the MMP for projective morphisms between complex analytic spaces, as developed in Fujino's previous works ([F5], [F6]).
• Avoidance of Extension Theorems: While the existence of flips in the MMP ultimately relies on extension theorems, Fujino argues that using the established MMP results (flips, divisorial contractions, basepoint-free theorems) is more accessible and conceptually transparent than invoking the specific, highly technical extension theorems used in [HM2].
• Key Technical Tools:
  • Kodaira Dimension Inequalities: Utilizing Theorem 3.1 (based on Campana's twisted weak positivity) to establish lower bounds on Kodaira dimensions.
  • Hacon–McKernan Criterion: Adapting the criterion for rational chain connectedness (Lemma 4.2 and Proposition 4.1) to the analytic setting.
  • MMP on Fibers: Running a $(K_Y + \Gamma)$-MMP over the base to reduce the problem to cases where the Kodaira dimension is non-positive or where specific contraction properties hold.
  • Adjunction and Discrepancies: Careful analysis of discrepancies and non-klt loci during the MMP process to track the "bad" locus (where rational connectedness might fail).

3. Key Contributions

The paper makes several significant technical contributions:

1. Extension to Complex Analytic Spaces: It successfully extends the Hacon–McKernan rational chain connectedness theorem to projective morphisms between complex analytic spaces, a setting where many algebraic tools do not directly apply.
2. Alternative Proof Strategy: It provides a proof that circumvents the direct use of the intricate extension theorems found in [HM1]. Instead, it relies on the Basepoint-Free Theorem and the existence of flips/contractions in the analytic MMP.
3. New Lemma (Lemma 5.2): A crucial new lemma is introduced which proves that irreducible components of the fiber are rationally chain connected modulo the non-klt locus by running an MMP on the fiber itself. This lemma replaces the difficult Kodaira dimension estimates used in the original algebraic proof.
4. Clarification of Dependencies: The paper clarifies that while the MMP itself depends on extension theorems (for the existence of flips), the application of the MMP to prove rational connectedness can be done using standard MMP arguments, making the result more accessible to readers familiar with the MMP but less so with the deepest extension results.

4. Main Results

Theorem 5.1 (Hacon–McKernan Rational Chain Connectedness Theorem - Analytic Version):
Let $f: X \to S$ be a projective morphism between normal complex varieties with $f_* \mathcal{O}_X \simeq \mathcal{O}_S$. Let $\Delta$ be an effective $\mathbb{Q}$-divisor such that $K_X + \Delta$ is $\mathbb{Q}$-Cartier, $\Delta$ is $f$-big, and $K_X + \Delta \sim_{\mathbb{Q}, f} 0$.
Then, for a point $s \in S$, after shrinking $S$, there exists a resolution $g: Y \to X$ such that the fiber $\pi^{-1}(s)$ (where $\pi = f \circ g$) is rationally chain connected modulo $g^{-1}(\text{Nklt}(X, \Delta))$.

• Note: If $(X, \Delta)$ is kawamata log terminal (klt), the fiber is rationally chain connected.

Theorem 1.1 (Main Application):
Let $X$ be a normal complex variety, $\Delta$ an effective $\mathbb{R}$-divisor with $K_X + \Delta$ $\mathbb{R}$-Cartier. Let $f: X \to S$ be a projective morphism.
Assume:

1. $-K_X$ is $f$-big.
2. $-(K_X + \Delta)$ is $f$-semiample.
   Then, for any bimeromorphic morphism $g: Y \to X$, the connected components of every fiber of the induced morphism $\pi: Y \to S$ are rationally chain connected modulo the inverse image of the non-klt locus of $(X, \Delta)$.

Corollaries:

• Corollary 1.2: For a divisorial log terminal (dlt) pair $(X, \Delta)$, the fibers of any bimeromorphic morphism are rationally chain connected. If $X$ is projective, $X$ is rationally chain connected iff it is rationally connected.
• Corollary 1.3: For a meromorphic map $f: X \dashrightarrow Y$ between normal complex varieties with $(X, \Delta)$ dlt, the image of the indeterminacy locus of $f$ in $Y$ is rationally chain connected.
• Corollary 1.4: If $f: X \dashrightarrow Y$ is a meromorphic map between complex analytic spaces with $(X, \Delta)$ dlt, and $Y$ contains no rational curves, then $f$ is a morphism.
• Theorem 1.7: Generalizes rational chain connectedness to quasi-log complex analytic spaces where $-\omega$ is ample.

5. Significance

• Accessibility: By avoiding the direct invocation of the highly technical extension theorems of Hacon–McKernan, Fujino makes the proof of rational chain connectedness more accessible to researchers familiar with the Minimal Model Program.
• Analytic Geometry: It bridges a gap between algebraic geometry and complex analytic geometry, confirming that deep structural results regarding rational connectedness hold in the analytic category. This is crucial for the study of Kähler manifolds and complex analytic spaces which are not necessarily algebraic.
• Foundational for MMP: The paper reinforces the utility of the MMP in complex analytic spaces (as developed in [F5], [F6]), demonstrating that it is a robust framework capable of handling problems previously thought to require specific algebraic extension techniques.
• Singularities: The results provide strong structural information about the fibers of resolutions of singularities for kawamata log terminal singularities, showing they are "small" in the sense of rational connectedness.

In summary, Fujino successfully adapts a major theorem in birational geometry to the complex analytic setting by leveraging the modern MMP framework, offering a cleaner, more modular proof that avoids the most opaque technical hurdles of the original algebraic proof.