Smooth Fibration Research Articles

Great circle fibrations and contact structures on the 3-sphere

Given any smooth fibration of the unit 3-sphere by great circles, we show that the distribution of 2-planes orthogonal to the great circle fibres is a tight contact structure, a fact well known in the special case of the Hopf fibrations. The proof expresses hypothesis and conclusion as differential inequalities involving functions on disks transverse to the fibres, and shows that one inequality implies the other.

Fano 4-folds with a small contraction

Let X be a smooth complex Fano 4-fold. We show that if X has a small elementary contraction, then ρX≤12, where ρX is the Picard number of X. This result is based on a careful study of the geometry of X, on which we give a lot of information. We also show that in the boundary case ρX=12 an open subset of X has a smooth fibration with fiber P1. Together with previous results, this implies that if X is a Fano 4-fold with ρX≥13, then every elementary contraction of X is divisorial and sends a divisor to a surface. The proof is based on birational geometry and the study of families of rational curves. More precisely the main tools are: the study of families of lines in Fano 4-folds and the construction of divisors covered by lines, a detailed study of fixed prime divisors, the properties of the faces of the effective cone, and a detailed study of rational contractions of fiber type.

Extending rationally connected fibrations from ample subvarieties

Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.

ON PROJECTIVE MANIFOLDS WITH PSEUDO-EFFECTIVE TANGENT BUNDLE

AbstractIn this paper, we develop the theory of singular Hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold X with pseudo-effective tangent bundle; X admits a smooth fibration $X \to Y$ to a flat projective manifold Y such that its general fibre is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general nonminimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

Optimal Symplectic Connections on Holomorphic Submersions

The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, which we call the optimal symplectic connection equation.We begin with a smooth fibration for which all fibres admit a constant scalar curvature Kähler metric. When the fibres admit automorphisms, such metrics are not unique in general, but rather are unique up to the action of the automorphism group of each fibre. We define an equation which, at least conjecturally, determines a canonical choice of constant scalar curvature Kähler metric on each fibre. When the fibration is a projective bundle, this equation specialises to asking that the hermitian metric determining the fibrewise Fubini‐Study metric is Hermite‐Einstein.Assuming the existence of an optimal symplectic connection and the existence of an appropriate twisted extremal metric on the base of the fibration, we show that the total space of the fibration itself admits an extremal metric for certain polarisations making the fibres small. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Contact structures induced by skew fibrations of R3

A smooth fibration of $\mathbb{R}^3$ by oriented lines is given by a smooth unit vector field $V$ on $\mathbb{R}^3$, for which all of the integral curves are oriented lines. Such a fibration is called skew if no two fibers are parallel, and it is called nondegenerate if $\nabla V$ vanishes only in the direction of $V$. Nondegeneracy is a form of local skewness, though in fact any nondegenerate fibration is globally skew. Nondegenerate and skew fibrations have each been recently studied, from both geometric and topological perspectives, in part due to their close relationship with great circle fibrations of $S^3$. Any fibration of $\mathbb{R}^3$ by oriented lines induces a plane field on $\mathbb{R}^3$, obtained by taking the orthogonal plane to the unique line through each point. We show that the plane field induced by any nondegenerate fibration is a tight contact structure. For contactness we require a new characterization of nondegenerate fibrations, whereas the proof of tightness employs a recent result of Etnyre, Komendarczyk, and Massot on tightness in contact metric 3-manifolds. We conclude with some examples which highlight relationships among great circle fibrations, nondegenerate fibrations, skew fibrations, and the contact structures associated to fibrations.

Relative singular support and the semi-continuity of characteristic cycles for étale sheaves

Recently, the singular support and the characteristic cycle of an etale sheaf on a smooth variety over a perfect field are constructed by Beilinson and Saito, respectively. In this article, we extend the singular support to a relative situation. As an application, we prove the generic constancy for singular supports and characteristic cycles of etale sheaves on a smooth fibration. Meanwhile, we show the failure of the lower semi-continuity of characteristic cycles in a higher relative dimension case, which is different from Deligne and Laumon’s result in the relative curve case.

Gluing formula of real analytic torsion forms and adiabatic limit

In this paper we use the adiabatic limit method to prove the gluing formula of real analytic torsion forms for a flat vector bundle on a smooth fibration under the assumption that the fiberwise twisted cohomology groups of the gluing hypersurface are vanished. In this paper we assume that the metrics have product structures near the gluing hypersurface.

On the Gluing Formula of Real Analytic Torsion Forms

In this paper we extend first the Bismut-Lott's analytic torsion form for flat vector bundles to the boundary case, then we establish its gluing formula on a smooth fibration under the assumption that a fiberwise Morse function exists. We assume that the metrics have product structures near the cutting hypersurface.

Higher spectral flow and an entire bivariant JLO cocycle

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

On the analyticity of the bivariant JLO cocycle

The goal of this note is to outline a proof that, for any l $\geq 0$, the JLO bivariant cocycle associated with a family of Dirac type operators along a smooth fibration $M\to B$ over the pair of algebras $(C^\infty (M), C^\infty(B))$, is entire when we endow $C^\infty(M)$ with the $C^{l+1}$ topology and $C^\infty(B)$ with the $C^{l}$ topology. As a corollary, we deduce that this cocycle is analytic when we consider the Frechet smooth topologies on $C^\infty(M)$ and $C^\infty(B)$.

Global smooth fibrations of ℝ3 by oriented lines

A smooth fibration of ℝ 3 by oriented lines is given by a smooth unit vector field V on ℝ 3 all of whose integral curves are straight lines. Such a fibration is said to be nondegenerate if dV vanishes only in the direction of V. Let ℒ be the space of oriented lines of ℝ 3 endowed with its canonical pseudo-Riemannian neutral metric. We characterize the nondegenerate smooth fibrations of ℝ 3 by oriented lines as the closed (in the relative topology) definite connected surfaces in ℒ. In particular, local conditions on ℒ imply the existence of a global fibration. Besides, for any such fibration the base space is diffeomorphic to the open disc and the directions of the fibers form an open convex set of the two-sphere. We characterize as well, in a similar way, the smooth (possibly degenerate) fibrations.

On the moduli of surfaces admitting genus two fibrations over elliptic curves

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that a surface admitting a smooth fibration as above is elliptic, and we employ results on the moduli of polarized elliptic surfaces to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes \({\mathcal{H}}(1,X(d), n)\) of morphisms of degree n from elliptic curves to the modular curve X(d), d ≥ 3. Ultimately, we show that the moduli spaces in the nonsmooth case are fiber spaces over the affine line \({\mathbb{A}}^1\) with fibers determined by the components of \({\mathcal{H}}(1,X(d), n)\).

The Obstruction to Fibering a Manifold over a Circle

Let M be a compact, connected, smooth manifold whose dimension is greater than five, and let / be a continuous map from M to the circle, which we denote by S. Suppose that / restricted to the boundary of M9 denoted by bM9 is a smooth fibration. We note that a map h from a smooth manifold N to S is a smooth fibration if h is smooth, and for each point x of N the derivative of h maps the tangent plane to N at x onto the tangent plane to S at h(x). We wish to address ourselves in this talk to the following problem.

The obstruction to fibering a manifold over a circle

Let M be a compact, connected, smooth manifold whose dimension is greater than five, and let / be a continuous map from M to the circle, which we denote by S. Suppose that / restricted to the boundary of M9 denoted by bM9 is a smooth fibration. We note that a map h from a smooth manifold N to S is a smooth fibration if h is smooth, and for each point x of N the derivative of h maps the tangent plane to N at x onto the tangent plane to S at h(x). We wish to address ourselves in this talk to the following problem.