Trivial Fibration Research Articles

Algebraic fibre spaces with strictly nef relative anti‐log canonical divisor

AbstractLet be a projective klt pair, and a fibration to a smooth projective variety with strictly nef relative anti‐log canonical divisor . We prove that is a locally trivial fibration with rationally connected fibres, and the base is a canonically polarized hyperbolic manifold. In particular, when is a single point, we establish that is rationally connected. Moreover, when and is strictly nef, we prove that is ample, which confirms the singular version of a conjecture by Campana and Peternell for threefolds.

A 2-categorical proof of Frobenius for fibrations defined from a generic point

AbstractConsider a locally cartesian closed category with an object $\mathbb{I}$ and a class of trivial fibrations, which admit sections and are stable under pushforward and retract as arrows. Define the fibrations to be those maps whose Leibniz exponential with the generic point of $\mathbb{I}$ defines a trivial fibration. Then the fibrations are also closed under pushforward.

Toric sheaves, stability and fibrations

For an equivariant reflexive sheaf over a polarised toric variety, we study slope stability of its reflexive pullback along a toric fibration. Examples of such fibrations include equivariant blow-ups and toric locally trivial fibrations. We show that stability (resp. unstability) is preserved under such pullbacks for so-called adiabatic polarisations. In the strictly semistable situation, under locally freeness assumptions, we provide a necessary and sufficient condition on the graded object to ensure stability of the pulled back sheaf. As applications, we provide various stable perturbations of semistable tangent sheaves, either by changing the polarisation, or by blowing-up a subvariety. Finally, our results apply uniformly in specific flat families and induce injective maps between the associated moduli spaces.

Thom property and Milnor–Lê fibration for analytic maps

AbstractLet (X, 0) be the germ of either a subanalytic set or a complex analytic space , and let be a ‐analytic map‐germ, with or , respectively. When , there is a well‐known topological locally trivial fibration associated with f, called the Milnor–Lê fibration of f, which is one of the main pillars in the study of singularities of maps and spaces. However, when that is not always the case. In this paper, we give conditions which guarantee that the image of f is well‐defined as a set‐germ, and that f admits a Milnor–Lê fibration. We also give conditions for f to have the Thom property. Finally, we apply our results to mixed function‐germs of type on a complex analytic surface with arbitrary singularity.

Zeros of One-Forms and Homologically Trivial Fibrations

Zeros of One-Forms and Homologically Trivial Fibrations

Cayley fibrations in the Bryant–Salamon Spin(7) manifold

On each complete asymptotically conical {{,mathrm{Spin},}}(7) manifold constructed by Bryant and Salamon, including the asymptotic cone, we consider a natural family of {{,mathrm{SU},}}(2) actions preserving the Cayley form. For each element of this family, we study the (possibly singular) invariant Cayley fibration, which we describe explicitly, if possible. These can be reckoned as generalizations of the trivial flat fibration of {mathbb {R}}^8 and the product of a line with the Harvey–Lawson coassociative fibration of {mathbb {R}}^7. The fibres will provide new examples of asymptotically conical Cayley submanifolds in the Bryant–Salamon manifolds of topology {mathbb {R}}^4, {mathbb {R}}times S^3 and {mathcal {O}}_{{mathbb {C}}{mathbb {P}}^1} (-1).

Collapsing to Alexandrov spaces with isolated mild singularities

Let Mj be a sequence of Riemannian manifolds with sectional curvature bound below collapsing to a compact Alexandrov space X of dimension k. Suppose that all but finitely many points of X are (k,δ)-strained and that the space of directions at each exceptional point contains k+1 directions making obtuse angles with each other. We prove that Mj admits a structure of locally trivial fibration over X for sufficiently large j. The same is true for collapsing sequences of Alexandrov spaces such that the infimum of the volume of the spaces of directions is sufficiently large relative to δ.

Realization of Fomenko-Zieschang invariants in closed symplectic manifolds with contact singularities

Abstract The topological bifurcations of Liouville foliations on invariant -manifolds that are induced by attaching toric -handles are investigated. It is shown that each marked molecule (Fomenko-Zieschang invariant) can be realized on an invariant submanifold of a closed symplectic manifold with contact singularities which is obtained by attaching toric -handles sequentially to a set of symplectic manifolds, while these latter have the structures of locally trivial fibrations over associated with atoms. Bibliography: 10 titles.

Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities

In this article we study the structure of klt projective varieties with nef anticanonical divisor (and more generally, varieties of semi-Fano type), especially the canonical fibrations associated to them. We show that The Albanese map for such variety is a locally constant fibration (that is, a locally trivial fibration such that X is isomorphic to the product of the universal cover of the Albanese torus by the fibre of the Albanese map quotient by a diagonal action of the fundamenatl group of the Albanese torus). If the smooth locus is simply connected, the MRC fibration of such variety is an everywhere defined morphism and induces a decomposition of this variety into a product of a rationally connected variety by a projective variety with trivial canonical divisor. These generalize the corresponding results for smooth projective varieties with nef anticanonical bundle in Cao (2019) and Cao-Höring (2019) to the klt case, and partially reduce the structure problem of these varieties to the singular Beauville-Bogomolov decomposition theorem proved by successive works of Greb-Kebekus-Peternell (2016), Druel (2018), Guenencia-Greb-Kebekus (2019) and Höring-Peternell (2019).

Systematics of the \u03b1\u2032 expansion in F-theory

Extracting reliable low-energy information from string compactifications notoriously requires a detailed understanding of the UV sensitivity of the corresponding effective field theories. Despite past efforts in computing perturbative string corrections to the tree-level action, neither a systematic approach nor a unified framework has emerged yet. We make progress in this direction, focusing on the moduli dependence of perturbative corrections to the 4D scalar potential of type IIB Calabi-Yau orientifold compactifications. We proceed by employing two strategies. First, we use two rescaling symmetries of type IIB string theory to infer the dependence of any perturbative correction on both the dilaton and the Calabi-Yau volume. Second, we use F/M-theory duality to conclude that KK reductions on elliptically-fibred Calabi-Yau fourfolds of the M-theory action at any order in the derivative expansion can only generate (α′)even corrections to the 4D scalar potential, which, moreover, all vanish for trivial fibrations. We finally give evidence that (α′)odd effects arise from integrating out KK and winding modes on the elliptic fibration and argue that the leading no-scale breaking effects at string tree-level arise from (α′)3 effects, modulo potential logarithmic corrections.

Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras

Let X be a complex manifold, π:E→X a locally trivial holomorphic fibration with fiber F, and (g,〈•,•〉) a Lie algebra with an invariant symmetric form. We associate to this data a holomorphic prefactorization algebra Fg,π on X in the formalism of Costello-Gwilliam. When X=C, g is simple, and F is a smooth affine variety, we extract from Fg,π a vertex algebra which is a vacuum module for the universal central extension of the Lie algebra g⊗H0(F,O)[z,z−1]. As a special case, when F is an algebraic torus (C⁎)n, we obtain a vertex algebra naturally associated to an (n+1)–toroidal algebra, generalizing the affine vacuum module.

Distinguishing open symplectic mapping tori via their wrapped Fukaya categories

In this paper, we present partial results towards a classification of symplectic mapping tori using dynamical properties of wrapped Fukaya categories. More precisely, we construct a symplectic manifold $T_\phi$ associated to a Weinstein domain $M$, and an exact, compactly supported symplectomorphism $\phi$. $T_\phi$ is another Weinstein domain and its contact boundary is independent of $\phi$. In this paper, we distinguish $\phi$ from $T_{1_M}$, under certain assumptions (Theorem 1.1). As an application, we obtain pairs of diffeomorphic Weinstein domains with the same contact boundary and whose symplectic cohomology groups are the same, as vector spaces, but that are different as Liouville domains. To our knowledge, this is the first example of such pairs that can be distinguished by their wrapped Fukaya category. Previously, we have suggested a categorical model $M_\phi$ for the wrapped Fukaya category $\mathcal{W}(T_\phi)$, and we have distinguished $M_\phi$ from the mapping torus category of the identity. In this paper, we prove $\mathcal{W}(T_\phi)$ and $M_\phi$ are derived equivalent (Theorem 1.9); hence, deducing the promised Theorem 1.1. Theorem 1.9 is of independent interest as it preludes an algebraic description of wrapped Fukaya categories of locally trivial symplectic fibrations as twisted tensor products.

Numerical characterization of some toric fiber bundles

Given a complex projective manifold X and a divisor D with normal crossings, we say that the logarithmic tangent bundle $$T_X(-\hbox {log}\, D)$$ is R-flat if its pull-back to the normalization of any rational curve contained in X is the trivial vector bundle. If moreover $$-(K_X+D)$$ is nef, then the log canonical divisor $$K_X+D$$ is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber F being a toric variety with boundary divisor $$D_{|F}$$ .

Positivity of the second exterior power of the tangent bundles

Let X be a smooth complex projective variety with nef ⋀2TX and dim⁡X≥3. We prove that, up to a finite étale cover X˜→X, the Albanese map X˜→Alb(X˜) is a locally trivial fibration whose fibers are isomorphic to a smooth Fano variety F with nef ⋀2TF. As a bi-product, we see that TX is nef or X is a Fano variety. Moreover we study a contraction of a KX-negative extremal ray φ:X→Y. In particular, we prove that X is isomorphic to the blow-up of a projective space at a point if φ is of birational type. We also prove that φ is a smooth morphism if φ is of fiber type. As a consequence, we give a structure theorem of varieties with nef ⋀2TX.

A fibration theorem for collapsing sequences of Alexandrov spaces

Suppose a sequence [Formula: see text] of Alexandrov spaces collapses to a space [Formula: see text] with only weak singularities. Yamaguchi constructed a map [Formula: see text] called an almost Lipschitz submersion for large [Formula: see text]. We prove that if [Formula: see text] has a uniform positive lower bound for the volumes of spaces of directions, which is sufficiently large compared to the weakness of singularities of [Formula: see text], then [Formula: see text] is a locally trivial fibration. Moreover, we show some properties on the intrinsic metric and the volume of the fibers of [Formula: see text].

The Bogomolov-Beauville-Yau decomposition for klt projective varieties with trivial first Chern class - without tears

We give a simplified proof (in characteristic zero) of the decomposition theorem for connected complex projective varieties with klt singularities and a numerically trivial canonical bundle. The proof mainly consists in reorganizing some of the partial results obtained by many authors and used in the previous proof but avoids those in positive characteristic by S. Druel. The single, to some extent new, contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibers without holomorphic vector fields. We first give the proof in the easier smooth case, following the same steps as in the general case, treated next. The last two words of the title are plagiarized from [4].

Bryant–Salamon [formula omitted] manifolds and coassociative fibrations

Bryant–Salamon constructed three 1-parameter families of complete manifolds with holonomy G2 which are asymptotically conical to a holonomy G2 cone. For each of these families, including their asymptotic cone, we construct a fibration by asymptotically conical and conically singular coassociative 4-folds. We show that these fibrations are natural generalizations of the following three well-known coassociative fibrations on R7: the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of ℂ3 with a line, and the Harvey–Lawson coassociative fibration. In particular, we describe coassociative fibrations of the bundle of anti-self-dual 2-forms over the 4-sphere S4, and the cone on ℂP3, whose smooth fibres are T∗S2, and whose singular fibres are R4∕{±1}. We relate these fibrations to hypersymplectic geometry, Donaldson’s work on Kovalev–Lefschetz fibrations, harmonic 1-forms and the Joyce–Karigiannis construction of holonomy G2 manifolds, and we construct vanishing cycles and associative “thimbles” for these fibrations.

Maps with finitely many critical points into high dimensional manifolds

Assume that there exists a smooth map between two closed manifolds $$M^m\rightarrow N^k$$ , where $$2\le k\le m\le 2k-1$$ , with only finitely many singular points, all of which are cone-like. If $$(m,k)\not \in \{(2,2), (4,3), (5,3), (8,5), (16,9)\}$$ , then $$M^m$$ admits a locally trivial topological fibration over $$N^k$$ and there exists a smooth map $$M^m\rightarrow N^k$$ with at most one critical point.

Deligne pairings and families of rank one local systems on algebraic curves

For smooth families $\mathcal{X} \to S$ of projective algebraic curves and holomorphic line bundles $\mathcal{L, M} \to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\langle \mathcal{L, M} \rangle \to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\mathcal{L}$ and $\mathcal{M}$. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.

Diffeomorphisms preserving Morse–Bott functions

Let f:M→R be a Morse–Bott function on a closed manifold M, so the set Σf of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by S(f)={h∈D(M)∣f∘h=h} the group of diffeomorphisms of M preserving f and let D(Σf) be the group of diffeomorphisms of Σf. We prove that the “restriction to Σf” map ρ:S(f)→D(Σf), ρ(h)=h|Σf, is a locally trivial fibration over its image ρ(S(f)).