On Kodaira fibrations with invariant cohomology

Bregman, Corey

doi:10.2140/gt.2021.25.2385

Cited by 6 publications

(5 citation statements)

References 21 publications

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“…An example of this phenomenon arises already in the case of doubly fibreed Kodaira fibrations (such as Atiyah and Kodaira's original examples). Moreover, there exist examples even of Kodaira fibrations that also admit pencils with multiple fibres (see [3]). But for the case of (non-Kähler) surface bundles the situation can be even more complex, as the following construction shows.…”

Section: Proofsmentioning

confidence: 99%

BNS Invariants and Algebraic Fibrations of Group Extensions

Friedl¹,

Vidussi

2021

J. Inst. Math. Jussieu

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Let G be a finitely generated group that can be written as an extension $$ \begin{align*} 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} \Gamma \longrightarrow 1 \end{align*} $$ where K is a finitely generated group. By a study of the Bieri–Neumann–Strebel (BNS) invariants we prove that if $b_1(G)> b_1(\Gamma ) > 0$ , then G algebraically fibres; that is, admits an epimorphism to $\Bbb {Z}$ with finitely generated kernel. An interesting case of this occurrence is when G is the fundamental group of a surface bundle over a surface $F \hookrightarrow X \rightarrow B$ with Albanese dimension $a(X) = 2$ . As an application, we show that if X has virtual Albanese dimension $va(X) = 2$ and base and fibre have genus greater that $1$ , G is noncoherent. This answers for a broad class of bundles a question of J. Hillman ([9, Question 11(4)]). Finally, we show that there exist surface bundles over a surface whose BNS invariants have a structure that differs from that of Kodaira fibrations, determined by T. Delzant.

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Section: Proofsmentioning

confidence: 99%

BNS Invariants and Algebraic Fibrations of Group Extensions

Friedl¹,

Vidussi

2021

J. Inst. Math. Jussieu

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“…An example of this phenomenon arises already in the case of doubly fibered Kodaira fibrations (such as Atiyah and Kodaira's original examples). Moreover there exist examples even of Kodaira fibrations which admit also pencils with multiple fibers (see [Br18]). But for the case of (non-Kähler) surface bundles the situation can be even more complex, as the following construction shows.…”

Section: Proofsmentioning

confidence: 99%

BNS Invariants and Algebraic Fibrations of Group Extensions

Friedl¹,

Vidussi²

2019

Preprint

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Let G be a finitely generated group that can be written as an extensionwhere K is a finitely generated group. By a study of the BNS invariants we prove that if b 1 (G) > b 1 (Γ) > 0, then G algebraically fibers, i.e. admits an epimorphism to Z with finitely generated kernel. An interesting case of this occurrence is when G is the fundamental group of a surface bundle over a surface F ֒→ X → B with Albanese dimension a(X) = 2. As an application, we show that if X has virtual Albanese dimension va(X) = 2 and base and fiber have genus greater that 1, G is noncoherent. This answers for a broad class of bundles a question of J. Hillman ([Hil15, Question 11(4)]).

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“…(3) Pencil kernels: In the case where X is a Kodaira fibration with excessive homology b 1 (G) − b 1 (Γ) equal to 2 or 4, Bregman proved in [Br18] the existence of an irregular pencil (in the sense of algebraic geometry) g : X → Σ where Σ is a surface of genus respectively 1 or 2. (For higher excessive homology the situation is unknown.)…”

Section: Subgroups and Virtual Homological Torsionmentioning

confidence: 99%