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Homology fiber bundles of varieties, that are not topological fiber bundles

This paper constructs flat, projective morphisms with smooth total spaces that are $\mathbb{Z}$ -homology fiber bundles but fail to be smooth, thereby disproving a conjecture by the second author and Fernández de Bobadilla.

Original authors: Maurício Corrêa, János Kollár

Published 2026-07-08

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CC BY 4.0

Original authors: Maurício Corrêa, János Kollár

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are looking at a stack of pancakes. In a perfect, smooth stack, every pancake is identical to the one above and below it. If you squish the stack slightly, the pancakes might change shape, but they all remain pancakes, and the way they connect to each other is consistent. In mathematics, this is called a fiber bundle. It's a way of saying, "If you look at any slice of this object, it looks exactly like every other slice."

For a long time, mathematicians wondered: If a stack of pancakes looks the same from a distance (in terms of its "holes" or "loops"), does that mean the pancakes are actually identical in shape?

This paper by Maurício Corrêa and János Kollár says: No, not necessarily. They built a mathematical "stack" that looks perfect from a distance but is actually broken and jagged up close.

Here is the breakdown of their discovery using simple analogies:

1. The Three Levels of "Sameness"

The paper discusses three ways to check if a stack of pancakes (a family of shapes) is "smooth" and uniform:

Level 1: The Smooth Check (The Real Thing). You look at the stack and see that every single pancake is perfectly round and smooth. This is the gold standard.
Level 2: The Shape Check (Homotopy). You look at the stack and ask, "Can I stretch or squish one pancake into the shape of another without tearing it?" If yes, they are "homotopy equivalent."
Level 3: The Hole Check (Homology). You look at the stack and just count the holes. "Does every pancake have exactly one hole? Two holes? Zero holes?" If the hole count is the same for every slice, they are "homology equivalent."

The Big Question: If a stack passes the "Hole Check" (Level 3), does it automatically pass the "Smooth Check" (Level 1)?

2. The Previous Belief (The Conjecture)

Before this paper, mathematicians had a strong hunch (a conjecture) that the answer was Yes. They thought:

If the hole counts are the same for every slice, the stack must be smooth and uniform.
They believed that if you found a stack where the hole counts were perfect but the stack was actually broken (singular), it would be impossible to construct.

3. The "Magic" Counter-Example

Corrêa and Kollár built a machine that creates a stack of pancakes that tricks the "Hole Check" but fails the "Smooth Check."

Here is how their machine works:

The Ingredients: They take a standard, slightly bumpy pancake (a mathematical shape with a few critical points) and a special, donut-shaped surface called an Abelian variety (think of it as a multi-dimensional torus or a donut).
The Twist: They take the donut and the bumpy pancake and twist them together. Imagine wrapping a ribbon around a bumpy ball. They do this in a very specific, repeating pattern (using a "rotation" and a "translation").
The Result: When they look at the final stack from a distance, the "bumps" cancel each other out perfectly. The "holes" in the math add up to zero. To the "Hole Check" inspector, the stack looks perfect. Every slice has the exact same number of holes.
The Catch: Up close, the stack is not smooth. The central slice is actually jagged and broken (singular). It's not a smooth pancake; it's a crumpled mess that just looks like a smooth one if you only count holes.

4. Why This Matters

This discovery is like finding a "magic trick" in geometry.

The Conjecture was Wrong: They disproved the idea that "Hole Check = Smoothness."
The Limit: They showed that you can have a "Homology Fiber Bundle" (a stack that passes the hole test) that is not a "Topological Fiber Bundle" (a stack that is actually smooth).
The "Donut" Factor: A key part of their trick was using the "donut" shape (Abelian variety). The donut has a special property: it can wrap around the bumps in the pancake and hide them. Without the donut, the bumps would be visible.

5. What They Didn't Do

It is important to note what this paper does not say:

They did not say this happens for every shape. They specifically built examples where the shapes have a specific type of "loop" (fundamental group) related to the number 2.
They did not say the shapes are "general" (a specific complex math term). Their shapes are special cases.
They did not claim this applies to physical objects in the real world (like actual pancakes or bridges). This is purely about abstract mathematical shapes.

Summary

The authors built a mathematical illusion. They created a family of shapes where every slice has the exact same number of holes (making them look identical to a "hole-counter"), but the slices are actually broken and jagged (not smooth). This proves that counting holes is not enough to guarantee that a shape is smooth and uniform. They solved a puzzle that mathematicians had been trying to crack for years, showing that the "Hole Check" is a weaker test than previously thought.

Technical Summary: Homology Fiber Bundles of Varieties That Are Not Topological Fiber Bundles

Problem Statement
The paper addresses a central question in equisingularity theory regarding the relationship between algebraic, analytic, and topological versions of fiber bundles. Specifically, it investigates the conditions under which a flat, projective morphism $f: Y \to \Delta$ (where $\Delta$ is a disc) that is a $\mathbb{Z}$ -homology fiber bundle is necessarily a topological fiber bundle.

This inquiry refines Conjecture 1 from [FdBK12], which posited the equivalence of three conditions for a projective morphism $f: Y \to \Delta$ from a complex analytic manifold:

$f$ is smooth (hence a differentiable fiber bundle).
$f$ is a homotopy fiber bundle.
$f$ is a $\mathbb{Z}$ -homology fiber bundle.

While the implications (1) $\Rightarrow$ (2) $\Rightarrow$ (3) are standard, and equivalence holds for fibers with isolated singularities or simply connected fibers (via Whitehead's theorem), the conjecture suggested that (3) might imply (1) in general. Previous work by Fernandez de Bobadilla [FdB05] and [FdBK12] provided examples where (2) holds but (1) fails, though in all known cases, the total space was singular. [FdBK12] conjectured that if the total space is smooth, then (3) implies (1).

Methodology
The authors construct counterexamples to the implication (3) $\Rightarrow$ (1) by creating flat, projective morphisms with smooth total spaces that are $\mathbb{Z}$ -homology fiber bundles but fail to be topological fiber bundles (and are not smooth).

The construction relies on a quotient technique involving an abelian variety and a variety with isolated singularities:

Setup: Let $X \to \Delta$ be a projective morphism with finitely many critical points $p_i$ over the origin. Let $\gamma$ be an automorphism of $X$ of finite order $c$ that commutes with the projection and fixes the critical points.
Quotient Construction: Let $A$ be an abelian variety and $\tau: A \to A$ a translation of order $c$ . The authors define the total space $Y$ as the quotient $(A \times X) / (\tau, \gamma)$ under the diagonal action. The projection $\pi_Y: Y \to \Delta$ is the induced morphism.
Homological Analysis: The core of the proof utilizes the Mayer-Vietoris sequence and the Leray spectral sequence. The authors analyze the cohomology of the fibers by decomposing them into regions near the singularities (Milnor fibers) and the smooth complement.
Key Lemma (Lemma 9): The authors prove that the projection $(A \times M)/(\tau, \gamma) \to A/(\tau)$ induces an isomorphism on cohomology if and only if the characteristic polynomial $\chi_T(x)$ of the monodromy action $T$ on the homology of the Milnor fiber $M$ satisfies $\chi_T(1) = \pm 1$ .
Example Generation: They construct specific families $X \to \Delta$ (using sums of powers in weighted projective spaces) where the monodromy on the Milnor fiber satisfies the condition $\chi(1) = \pm 1$ .

Key Results

Theorem 3: The authors prove that if the characteristic polynomial of the monodromy action on the Milnor fiber satisfies $\chi(1) = \pm 1$ , the resulting quotient family $Y \to \Delta$ is a $\mathbb{Z}$ -homology fiber bundle. However, $Y \to \Delta$ is neither smooth nor a topological fiber bundle.
Disproof of Conjecture 1: The constructed examples demonstrate that a smooth total space $Y$ can admit a $\mathbb{Z}$ -homology fiber bundle structure without being a topological fiber bundle. This disproves the conjecture that (3) implies (1) for smooth total spaces.
Properties of Fibers: In the constructed examples, the fibers $Y_t$ have a fundamental group $\mathbb{Z}_2^r$ (for $r \ge 1$ ). Consequently, $Y \to \Delta$ is never a homotopy fiber bundle, leaving the equivalence of (1) and (2) as an open question.
Specific Examples:
- Example 4: A family of surfaces where the special fiber is a cuspidal curve bundle over an elliptic curve quotient (not normal), and general fibers are hyperelliptic surfaces.
- Example 14: A family of threefolds with terminal singularities derived from the $E_8$ singularity ( $x^2 + y^3 + z^5 = 0$ ), where the Milnor monodromy is the Coxeter transformation of the $E_8$ lattice.

Significance and Claims
The paper claims to provide the first counterexamples to the conjecture that a smooth total space which is a $\mathbb{Z}$ -homology fiber bundle must be a topological fiber bundle. The authors emphasize that their examples are flat, projective, and possess smooth total spaces, yet fail to be topological fiber bundles.

The authors note several specific characteristics of their findings:

The property of being a $\mathbb{Z}$ -homology fiber bundle is not stable under finite étale covers; in their construction, the product $A \times X$ is not a $\mathbb{Z}$ -homology fiber bundle, but the quotient $Y$ is.
The fibers in their examples are never of general type, as they admit a finite étale cover that is a product with an abelian variety.
The construction relies on the topological property that abelian varieties are homeomorphic to a product of circles ( $S^1 \times N$ ).

The work leaves open the question of whether (1) and (2) are equivalent (i.e., whether a smooth morphism that is a homotopy fiber bundle must be smooth), as their examples fail to be homotopy fiber bundles. The authors suggest that future work could explore examples where the canonical class of the fibers is ample.