Fano Fibrations Overview

• Fano fibrations are projective morphisms f: X → Z where X is klt and -Kₓ is f-ample, serving as the relative version of classical Fano varieties.
• They play a crucial role in the Minimal Model Program and birational classification by underpinning boundedness results and moduli-theoretic constructions.
• Their structure drives advances in geometric analysis, mirror symmetry, and arithmetic, linking canonical bundle formulas and bounded complement theory.

A Fano fibration is a projective morphism between normal algebraic varieties with connected fibers such that the total space is klt and the anti-canonical divisor is relatively ample over the base. This class includes varieties appearing in central outputs of the Minimal Model Program, such as Mori fiber spaces, and provides a relative version of Fano varieties in the context of birational classification. Fano fibrations, as a prototype of the “Fano type fibration,” are also essential in boundedness problems, moduli theory, birational geometry (flips, divisorial contractions), higher-dimensional algebraic geometry, and are tightly connected to the structure of Calabi–Yau and log Calabi–Yau fibrations.

1. Definition and Fundamental Properties

Let $f: X \to Z$ be a projective morphism of normal varieties with %%%%1%%%%. The morphism $f$ is a Fano fibration if

• $X$ is kawamata log terminal (klt),
• $-K_X$ is $f$-ample, i.e., $-K_X$ is ample on all fibers,
• $K_X$ is $\mathbb{Q}$-Cartier.

The general fiber is thus a (possibly singular) Fano variety, and when $Z$ is a point one recovers absolute Fano varieties. Allowing for more flexibility, one considers Fano-type fibrations: $f: X \to Z$ is of Fano type if there exists some boundary $C \geq 0$ with $(X, C)$ klt and $-(K_X + C)$ ample over $Z$, equivalently, $-K_X$ is big over $Z$.

Many variants appear:

• Log Calabi–Yau fibration: $(X, B) \to Z$ such that $K_X + B \sim_\mathbb{R} 0/Z$ and $(X, B)$ is log canonical.
• Toric Fano fibration: A proper equivariant map $f: X \to Y$ between toric varieties with $-K_X$ $f$-ample (possibly with toric boundary).
• Relative Fano fibration: A flat, projective, Gorenstein morphism $f: X \to S$ with geometrically integral fibers, whose relative anticanonical bundle $-K_{X/S}$ is $f$-relatively ample (Martín et al., 2010).

2. Core Boundedness and Classification Results

Fano fibrations are at the heart of the relative version of the Borisov–Alexeev–Borisov (BAB) boundedness theorem. For fixed dimension $d$, degree bound $r$, and $\varepsilon>0$, the family of $(d, r, \varepsilon)$-Fano-type fibrations $(X, B) \to Z$ is bounded (Birkar, 2022). More precisely:

• The varieties $X$ underlying all such fibrations form a bounded family.
• If the coefficients of the boundary $B$ are contained in a finite set, pairs $(X, B)$ are log bounded.
• These results extend to generalized pairs $(X, B+M)$ with $M$ a nef moduli part.
• Effective bounds exist for the volume $\operatorname{vol}(\ell f^*A - K_X)$, birationality indices for adjoint linear systems, and quantitative control over singularities (e.g., positive lower bounds for lc-thresholds).

Central objects in the birational classification arise as Fano fibrations: Mori fiber spaces ($\rho(X/Z)=1$, $-K_X$ $f$-ample), flips, divisorial contractions, and certain crepant birational models.

3. Singularities, Canonical Bundle Formula, and Complements

Singularity theory interacts deeply with Fano fibrations via the canonical bundle formula and the theory of complements. Given a klt pair $(X, B)$ and $f: X \to Z$ with $K_X + B \sim_\mathbb{Q} f^*(K_Z + B_Z + M_Z)$, one obtains key boundedness and finiteness properties:

• There is $\delta>0$ such that the discriminant b-divisor $B_Z$ has coefficients $\le 1 - \delta$ and the base generalized pair $(Z, B_Z + M_Z)$ is generalized $\delta$-lc (Birkar, 2023).
• When $\dim Z = 1$, all multiplicities of fibers $f^*z$ are uniformly bounded, and relative-global $N$-complements (klt or lc) exist, with $N$ depending only on dimension, singularities, and coefficients (Choi et al., 2024).
• The canonical bundle formula underpins bounding singularities and moduli-theoretic applications.

Shokurov’s conjecture on bounded complements is confirmed for toric Fano fibrations with hyperstandard coefficients: there is a uniform $n$ such that $n$-complements exist, and the pair remains klt (Ambro, 2022).

4. Birational Geometry and Structure Theorems

Fano fibrations play a decisive role in the birational geometry of higher-dimensional varieties:

• Any projective variety is expected to be birational to either a canonically polarized variety, a Calabi–Yau fibration, or a Fano fibration (Birkar, 2022).
• In the context of Fano 4-folds, rational contractions of fiber type (rational fibrations) are structurally governed by the existence of movable divisors that are not big; such fibrations can be classified via special, quasi-elementary, or product structures, with explicit Picard number bounds (e.g., $\rho(X)\leq 18$ outside simple product cases) (Casagrande, 2019, Casagrande et al., 2024, Montero et al., 2018).
• In dimension four, new sharp bounds, special families, and the structure of non-product Fano 4-folds with large Picard numbers have been worked out in detail (Casagrande et al., 2024).
• The theory of Fourier–Mukai transforms for Fano fibrations indicates the extreme rigidity of derived categories in this context: relative autoequivalences of $D^b(X)$ are all of “trivial” geometric type (automorphism, line bundle twist, shift) (Martín et al., 2010).

5. Moduli and Arithmetic Considerations

Boundedness and the existence of complements for Fano fibrations have major implications for constructing moduli spaces:

• For any fixed dimension, singularities, and coefficient set, Fano fibrations admit a uniform complement and occupy a bounded family in moduli (Choi et al., 2024, Birkar, 2022).
• Relative-global complements offer strong rigidity, crucial for constructing moduli stacks with “finite type” properties.
• In the toric and test-configuration context, Fano fibrations naturally support moduli-theoretic compactifications compatible with K-stability frameworks; the existence of good moduli spaces for K-semistable Fano fibrations is expected (via higher $\Theta$-stratifications and the theory of weighted volumes) (Odaka, 17 Jun 2025).
• In arithmetic or function field settings, Fano fibrations over curves are deeply connected to Manin’s Conjecture and arithmetic bifurcations—counting sections reduces to comparing anticanonical degrees, input from moduli of sections, and distribution of “good” versus “accumulating” components (Lehmann et al., 2023).

6. Geometric Analysis, Mirror Symmetry, and Degeneration Theory

Fano fibrations are central to understanding degenerations in geometric analysis, moduli, and mirror symmetry:

• The continuity method and Kähler–Ricci flow on compact Kähler manifolds with non-nef $K_X$ produce in finite time a Fano fibration, with the collapsing limit described by singular Kähler metrics solving twisted Kähler–Einstein equations involving Weil–Petersson type forms on the base (Bednarek, 26 Dec 2025, Zhang et al., 2016).
• The base of the fibration supports a singular twisted Kähler–Einstein metric, with explicit curvature formulas and Chern class decompositions in submersion cases.
• In the non-compact analytic setting, Fano fibrations provide the algebro-geometric model underpinning Kähler–Ricci shrinkers, their moduli, and the translation between analytic “bubbling” and algebraic degenerations (notably via the minimization of weighted volumes among valuations centered in fibers) (Sun et al., 2024, Odaka, 17 Jun 2025).
• Singular SYZ (Strominger–Yau–Zaslow) torus fibrations on Fano threefolds have been constructed, revealing explicit connections to affine manifolds, Lagrangian geometry, and compatibility with mirror symmetry (e.g., cohomological and intersection-theoretic invariants match the theoretical Fano mirror partners) (Prince, 2018).
• The classification of Fano fibrations in the toric case involves a fine combinatorial structure (fans, polytopes, moment polyhedra), extensive use of geometry-of-numbers techniques, and shows that minimal log discrepancies and $n$-complement indices fall into finite sets (Ambro, 2022).

7. Role in Birational Conjectures and Open Problems

A variety of birational and moduli-theoretic conjectures are deeply entangled with properties of Fano fibrations:

• The boundedness conjecture, asserting that Fano varieties and Fano fibrations with fixed local singularity type and other invariants form a bounded family, is established in many settings and underpins progress on moduli and effective birational geometry (Birkar, 2022, Jiang, 2015).
• Conjectures on the boundedness of fiber multiplicities (over curves or higher base) reduce, through the canonical bundle formula and appropriate log-lc thresholds, to conditions on the boundedness of singularities, the base, and (generalized) Calabi–Yau boundedness (Chen et al., 2022, Birkar, 2023).
• The Shokurov–McKernan conjecture on singularities of the base of Fano-type fibrations is proved: there is a uniform lower bound $\delta>0$ for the log canonicity of the induced generalized pair on the base (Birkar, 2023).
• Structure theorems for varieties admitting int-amplified endomorphisms ensure, up to quasi-étale cover, that such varieties are of Fano type over an abelian variety; in the rationally connected case, the structure lifts to $X$ itself (2002.01257).

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Summary Table: Key Concepts in Fano Fibrations

Concept	Definition/Property	Reference
Fano fibration	$f: X \to Z$ with $X$ klt, $-K_X$ $f$-ample	(Birkar, 2022)
Fano type fibration	$-K_X$ big over $Z$ or $-(K_X+B)$ ample	(Birkar, 2022, Birkar, 2023)
Boundedness	Existence of bounded family for $(d, r, \epsilon)$	(Birkar, 2022, Jiang, 2015)
Bounded complements	Uniform $N$ with relative-global $N$-complement	(Choi et al., 2024, Ambro, 2022)
Relative Fourier–Mukai group	Trivial: automorphisms, twist, shift	(Martín et al., 2010)
Canonical bundle formula	$K_X + B \sim_\mathbb{Q} f^*(K_Z+B_Z+M_Z)$	(Birkar, 2023, Chen et al., 2022)
Lefschetz defect ($\delta_X$)	Maximal codimension of $N_1(D,X) \subset N_1(X)$	(Montero et al., 2018, Casagrande et al., 2024)

One finds that Fano fibrations anchor the structure of birational, arithmetic, and geometric-analytic classification in higher-dimensional algebraic geometry, giving rise to explicit moduli theories, control of singularities and degenerations, and robust applications in both classical and modern research directions.