Invisible singularities in complex algebraic geometry

=== DRAFT ===
=== DRAFT ===
Technical Summary: Invisible Singularities in Complex Algebraic Geometry

Problem Statement
The paper addresses a central question in equisingularity theory: the relationship between algebraic, analytic, and topological notions of singularity. While Zariski's equisingularity concepts were expected to coincide in higher dimensions, previous counterexamples by Fernández de Bobadilla involved total spaces that were singular. A prevailing belief, supported by results in dimensions 2 and 3, was that if $Y$ is a smooth complex projective variety and $g: Y \to C$ is a proper morphism to a curve, then the topological structure of $Y$ must detect the presence of critical points (singular fibers). Specifically, it was conjectured that if the morphism looks topologically smooth (e.g., is a homotopy fiber bundle or has constant cohomology sheaves), it must be algebraically smooth (a submersion).

The authors investigate whether "invisible" singularities exist: morphisms $g: Y \to \mathbb{P}^1$ from a smooth variety $Y$ that possess singular fibers yet appear topologically smooth. This inquiry targets four specific open problems:

1. The smoothness conjecture of Fernández de Bobadilla and Kollár [FdBK12].
2. A question by Kollár and Pardon [KP12] regarding the homotopy type of universal covers of varieties fibered over curves.
3. Kotschick's conjecture [Kot22] on the equivalence of admitting a holomorphic one-form without zeros versus a real closed one-form without zeros.
4. A conjecture by Schreieder [Sch21] concerning the exactness of Aomoto complexes and the existence of non-vanishing forms.

Methodology
The authors construct explicit counterexamples using a combination of quotient constructions, blow-ups, and topological smoothing techniques.

1. Quotient Construction (Local Model): The core construction begins with a projective morphism $\pi: Z \to \Delta$ (over a disc) having isolated critical points. By taking a quotient of $Z \times A$ (where $A$ is an abelian variety) by a diagonal action of a finite cyclic group, the authors create a new morphism $\pi_Y: Y \to \Delta$. By carefully selecting the automorphism $\gamma$ of $Z$ and the translation $\tau$ of $A$ such that the characteristic polynomial of the monodromy action on the Milnor fiber satisfies $\chi(1) = \pm 1$, they ensure the resulting map is a $\mathbb{Z}$-homology fiber bundle but not a homotopy fiber bundle (Section 2).

2. Blow-up and Simply Connected Fibers: To achieve simply connected fibers (a requirement for the global theorems), the authors embed the singular quotient $Y$ into a smooth projective bundle $P$ with simply connected fibers and perform a blow-up $X = \text{Bl}_Y P$. Using Proposition 17, they show that the induced map $X \to \Delta$ remains a $\mathbb{Z}$-homology fiber bundle but is not a submersion. Since the fibers are simply connected, Whitehead's theorem implies $X \to \Delta$ is a homotopy fiber bundle (Section 3).

3. Globalization and Smoothing: The local examples are globalized to morphisms $g: Y \to \mathbb{P}^1$. To prove that these maps "look" topologically smooth, the authors employ Smale's $h$-cobordism theorem and Cerf's pseudo-isotopy theorem (Section 4). They demonstrate that for morphisms to Riemann surfaces with simply connected fibers, if the map is a $\mathbb{Z}$-homology fiber bundle, it is homotopic to a $C^\infty$-submersion that agrees with the original map outside a small neighborhood of the singular fibers.

4. Applications to Specific Conjectures:

   • Kotschick's Conjecture: By taking a fiber product with an elliptic curve, they construct a variety $X$ where the Albanese map is a homotopy fiber bundle (implying the existence of non-vanishing real forms) but not a submersion (implying all holomorphic forms have zeros).
   • Aomoto Complexes: Using a rational cohomology torus construction based on Debarre–Jiang–Lahoz [DJL17], they build a variety where the Aomoto complex is exact for all finite étale covers, yet no real closed one-form is non-vanishing.

Key Results

• Theorem 1: There exist morphisms $g: Y \to \mathbb{P}^1$ from smooth complex projective varieties $Y$ (specifically of dimension 5) such that:

  • $g$ has singular fibers.
  • $R^i g_* \mathbb{Z}_Y$ are constant sheaves (topologically constant cohomology).
  • $g$ is a homotopy fiber bundle with simply connected fibers.
  • All fibers are PL-manifolds, but $g$ is not a $C^0$-fiber bundle.
  • $g$ is homotopic to a locally trivial $C^\infty$-fiber bundle that agrees with $g$ away from the singular fibers.
  • Significance: This disproves the Fernández de Bobadilla–Kollár smoothness conjecture.

• Corollary 3 (Kollár–Pardon & Kotschick): There exists a smooth projective variety $X$ (dimension 5) such that:

  • The universal cover $\tilde{X}$ has the homotopy type of a finite CW complex, yet the Albanese map is not smooth.
  • Every nonzero class in $H^1(X, \mathbb{R})$ can be represented by a non-vanishing real closed one-form, but the harmonic representative of every such class has zeros. Consequently, $X$ admits no holomorphic one-form without zeros.
  • Significance: This provides negative answers to Kollár–Pardon's question on universal covers and Kotschick's conjecture.

• Theorem 4: There exists a smooth projective variety $X$ such that for every finite connected étale cover $\pi: X' \to X$ and every nonzero $\omega \in H^0(X', \Omega^1_{X'})$, the Aomoto complex is exact, yet every real closed one-form on $X$ has a zero.

  • Significance: This disproves Schreieder's conjecture that exactness of the Aomoto complex implies the existence of a non-vanishing real form.

• Theorem 40: Counterexamples to the smoothness conjecture with simply connected fibers do not exist in dimension 3. If $Y$ is a smooth complex projective threefold and $g: Y \to \mathbb{P}^1$ is a morphism with simply connected fibers that is a $\mathbb{Q}$-homology fiber bundle, then $g$ is smooth.

  • Significance: This establishes a sharp dimensional boundary for the phenomenon of invisible singularities.

Significance and Claims
The paper claims to provide the first examples of "invisible singularities" where the total space is smooth, yet the morphism possesses singular fibers that are undetectable by standard topological invariants (homology, homotopy type, and even $C^\infty$-smoothing up to homotopy).

The authors emphasize that their construction is explicit and simple, with the smallest example occurring in dimension 5, where the critical locus of the morphism has dimension 1. They clarify that while the general construction allows for higher-dimensional critical loci, the specific counterexample to the smoothness conjecture presented in Theorem 1 features a 1-dimensional critical locus. The work resolves long-standing questions in complex geometry by demonstrating that algebraic smoothness is not a topological invariant in dimensions $\geq 4$, even under strong topological constraints like simply connected fibers. The paper remains modest regarding dimension 4, noting that the question remains open in that dimension.

A scientific accuracy reviewer checked the draft against the paper and flagged these problems:

• Summary claims the phenomenon relies on a critical locus of dimension 1, but the paper states the smallest example has a 1D critical locus, while the general construction allows higher dimensions.

Produce a corrected version of the draft. Fix ONLY what the reviewer flagged (verify each point against the paper) and keep everything else — the structure, the level of detail, the terminology — unchanged. Output ONLY the corrected summary.