Kähler Mapping Torus in Complex Geometry

• Kähler Mapping Tori are constructed by quotienting a compact Kähler manifold via an automorphism to form a fiber bundle over S¹ while preserving the Kähler structure.
• In complex dimension three, these tori are classified up to finite étale covers and bimeromorphic equivalence, with model types including ℙ¹-, ℙ²-, and Hirzebruch surface bundles.
• The persistence of zero-free holomorphic 1-forms and finite monodromy illustrates deep connections between topology, complex analytical structure, and the Kähler minimal model program.

A Kähler mapping torus is a construction that produces a complex manifold as the quotient of a Kähler manifold by the action of an automorphism, typically realized as a fiber bundle over the circle whose fiber is a compact Kähler manifold. These objects serve as key examples in differential and complex geometry, illuminating deep relations between topology, complex analytic structure, and global geometric properties such as the existence of holomorphic 1-forms without zeros. In dimension three, Kähler mapping tori are fully classified up to finite étale cover and bimeromorphic equivalence, enabling a structural understanding of compact Kähler threefolds that fiber smoothly over $S^1$ (Pietig, 27 Jun 2025).

1. Definition and Construction of Kähler Mapping Tori

Let $F$ be a compact Kähler surface, and let $\varphi: F \to F$ be a smooth diffeomorphism or bimeromorphic automorphism. The Kähler mapping torus $M_\varphi$ is defined as

$$M_\varphi = (F \times [0,1])\,/\,(x,1) \sim (\varphi(x),0)$$

or equivalently as the quotient $(F \times \mathbb{R})/\mathbb{Z}$, where $n \in \mathbb{Z}$ acts via $n \cdot (x,t) = (\varphi^n(x), t-n)$. There is a natural projection $M_\varphi \to S^1 = \mathbb{R}/\mathbb{Z}$, making $M_\varphi$ a $C^\infty$-fiber bundle over $S^1$ with fiber $F$.

A mapping torus $M_\varphi$ is said to "carry a Kähler structure" if its total space admits a complex-analytic structure and a real closed $(1,1)$-form $\omega$ that is everywhere positive, thus making $M_\varphi$ itself a compact Kähler manifold. The existence of a nowhere-vanishing holomorphic 1-form $\Omega \in H^0(M_\varphi, \Omega^1)$, by Tischler's theorem and Hodge theory, forces the smooth fibration $M_\varphi \to S^1$ and requires that $\varphi$ preserve the Kähler class of $F$ (Pietig, 27 Jun 2025).

2. Main Classification in Complex Dimension Three

Let $X$ be a smooth compact connected Kähler threefold admitting a $C^\infty$-fiber bundle structure $X \to S^1$. The main result provides a classification of such $X$ up to finite étale cover and bimeromorphic equivalence. There exists:

• a finite étale cover $\tau: \widetilde{X} \to X$,
• a compact Kähler base $B$,
• a complex torus $A = \mathbb{C}^g/\Lambda$ of $g>0$,
• a cohomology class $\eta \in H^1(B, \mathcal{O}_B^\oplus g/\Lambda)$ with torsion Chern class $c(\eta) \in H^2(B, \Lambda)$, such that $\widetilde{X}$ is bimeromorphic to a fiber bundle $\pi_\eta: (B \times A)^\eta \to B$ whose fibers are either $\mathbb{P}^1$, $\mathbb{P}^2$, or a Hirzebruch surface $$F_n = \mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$$.

This structure is reflected in a diagram: $$\Sigma = \mathbb{P}^k \text{ or Hirzebruch} \to \widetilde{X} \to T \to B$$ where $T \to B$ is a locally trivial torus bundle (trivial monodromy after a finite étale cover), and $\widetilde{X} \to T$ is an isomorphism or a $\mathbb{P}^1$-, $\mathbb{P}^2$-, or $F_n$-bundle (Pietig, 27 Jun 2025).

3. Holomorphic 1-Forms Without Zeros and Monodromy Finiteness

A holomorphic 1-form $\Omega \in H^0(X, \Omega_X^1)$ is zero-free if and only if the complex

$$\cdots \to H^{i-1}(X, \mathbb{C}) \xrightarrow{\wedge \Omega} H^i(X, \mathbb{C}) \xrightarrow{\wedge \Omega} H^{i+1}(X, \mathbb{C}) \to \cdots$$

is exact. The theorem of Kotschick and Schreieder establishes that exactness for all finite étale covers is equivalent to the existence of a nowhere-vanishing real closed 1-form, or equivalently, to $X$ admitting a $C^\infty$-fibration over $S^1$ (Pietig, 27 Jun 2025).

The proof of the classification uses the property that the pullback of $\Omega$ to any finite étale cover remains nontrivial on each torus fiber or exceptional divisor. Exactness ensures that the monodromy of any torus- or elliptic-fibration arising in the minimal or Iitaka fibration is finite, becoming trivial after suitable étale covering. This reduction produces an isotrivial torus bundle or an equivariant Weierstrass model with invariant Hodge structure, making the total space a torsor for a trivial torus bundle.

4. Reduction Steps and Structure via the Minimal Model Program

The classification proceeds via:

• Reduction to a minimal or Mori fiber space using the Kähler Minimal Model Program (MMP) and blow-downs along elliptic curves.
• A case-by-case analysis classified by the Kodaira dimension $\kappa(X)$:
  • If $\kappa=3$ (general type), such $X$ cannot occur by Chern number considerations.
  • If $\kappa=2$, $X$ is an elliptic fibration over a surface; after étale base change and trivialization of monodromy, $X$ is bimeromorphic to $(S \times E)^\eta$.
  • If $\kappa=1$, $X \to C$ is a torus or elliptic bundle over a base curve; a finite étale cover and monodromy trivialization produces a structure $(C' \times A)^\eta$.
  • If $\kappa=0$, Beauville–Bogomolov decomposition gives $X' \cong Y \times A$.
  • If $\kappa=-\infty$, after reduction and étale covering, $X$ is a $\mathbb{P}^1$-, $\mathbb{P}^2$-, or $F_n$-bundle over a torus-bundle base.

In all cases, the Hodge-theoretic exactness of $\wedge \Omega$ guarantees that holomorphic 1-forms remain nonzero along the appropriate factors of the structure (Pietig, 27 Jun 2025).

5. Kotschick's Conjecture in Dimension Three

Kotschick's conjecture asserts that for a compact Kähler manifold $X$, the existence of a zero-free holomorphic 1-form is equivalent to the existence of a closed, nowhere-vanishing real 1-form (i.e., a smooth fibration over $S^1$). The implication from real to holomorphic (the subtle direction) is established in dimension three: after finite étale cover and resolving blow-downs, any $X$ that fibers over $S^1$ becomes bimeromorphic to a bundle of the classified type. A nonzero holomorphic 1-form from the torus or elliptic curve factor extends to the total space, including across exceptional divisors, thus ensuring the existence of a zero-free holomorphic 1-form (Pietig, 27 Jun 2025).

6. Model Bundles: $\mathbb{P}^1$-, $\mathbb{P}^2$-, and Hirzebruch Surface-Bundles

All Kähler mapping tori in dimension three, upon passing to a finite étale cover and resolving singularities, are bimeromorphic to one of three model types:

• $\mathbb{P}^1$-Bundles: e.g., $E \times \mathbb{P}^1 \to E$ for an elliptic curve $E$, or nontrivial examples formed as the mapping torus of a Hirzebruch surface automorphism with finite-order monodromy.
• $\mathbb{P}^2$-Bundles: e.g., $\mathbb{P}^2 \times E$ with the Kähler metric induced from the product, or more generally, bundles with projective linear monodromy that reduces to the product case after finite covering.
• Hirzebruch Surface ($F_n$)-Bundles: twisted torus bundles of the form $(Y \times E')^\eta \to Y$, with $Y$ a Kähler base and extension class $\eta$. For $n>0$, automorphisms of $F_n$ covering base translations yield mapping tori that trivialize upon passing to a finite étale cover.

Key features common to these examples are the trivialization (or finiteness) of monodromy ensuring the preservation of Kähler structure after covering, the persistence of zero-free holomorphic 1-forms pulled up from the torus/elliptic factor, and computability of Chern classes from standard models (Pietig, 27 Jun 2025).

Model Bundle Types and Fiber Geometry

Bundle Type	Fiber	Holomorphic 1-Form Source
$\mathbb{P}^1$	$\mathbb{P}^1$ or $F_n$	Torus/elliptic factor
$\mathbb{P}^2$	$\mathbb{P}^2$	Torus/elliptic factor
Hirzebruch surface	$F_n$	Torus/elliptic factor

A plausible implication is that all smooth compact connected Kähler threefolds with a $C^\infty$-bundle structure over $S^1$ decompose, up to finite étale cover and birational transformations, into these three explicit bundle types.

7. Significance and Open Directions

The classification of Kähler mapping tori in dimension three not only settles Kotschick's conjecture in this setting, but also serves as a blueprint for analyzing the topology, birational geometry, and holomorphic forms on higher-dimensional Kähler manifolds fibering over $S^1$. The explicit description in terms of classical surface bundles and torus bundles clarifies the role of finite monodromy in preserving Kähler geometry under mapping torus construction. The persistence of holomorphic 1-forms without zeros, controlled by Hodge-theoretic exactness, suggests links to other rigidity phenomena and obstructions in higher dimensions.

Further investigation may address the higher-dimensional analogs of the classification, the interaction of mapping torus structures with various minimal model programs, and refined invariants such as variation of Hodge structure and derived categories in the mapping torus context.