The paper studies algebraic fibre spaces with specific properties and proves key results about their structure.
problem Analyzing properties of algebraic fibre spaces with strictly nef relative anti-log canonical divisors.
method Using projective klt pairs and fibration techniques, the paper proves locally constant fibration properties and rational connectedness.
result The fibration is locally constant with rationally connected fibers, and the base is a canonically polarized hyperbolic projective manifold.
The paper studies Kähler manifolds with partially semi-positive curvature and rational connectedness.
problem Analyzing compact Kähler manifolds with partially semi-positive curvature and rational connectedness.
method Proving rational connectedness for manifolds with BC-p positive tangent bundles, and applying these results to curvature conditions. result Confirming a conjecture and generalizing results on rational connectedness and curvature conditions.
Helix surfaces in Anti-de Sitter space maintain constant Gaussian curvature.
problem Understanding helix surfaces in Anti-de Sitter space with Berger-like metrics.
method Proved helix surfaces have constant Gaussian curvature and described them explicitly.
result Explicit local description of helix surfaces in terms of isometries and curves.
The Hopf fibration is rigid among minimal maps between spheres.
problem Characterizing minimal submersions between spheres.
method Analyzing the properties of the Hopf fibration and minimal maps.
result The Hopf fibration is the only minimal submersion from S3 to S2 under certain conditions. Study on conditions for singular local tube fibrations.
problem Conditions for singular local tube fibrations.
method Analyzes the most general condition for singular local tube fibrations.
result Provides conditions for the existence of singular local tube fibrations.
Heinz Hopf's famous fibrations of the 2n+1-sphere by great circles, the 4n+3-sphere by great 3-spheres, and the 15-sphere by great 7-spheres have a number of interesting properties. Besides providing the first examples of homotopically nontrivial maps from one sphere to another sphere of lower dimension, they all share…
Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.
problem Compact Kähler manifolds with pseudo-effective tangent bundles.
method Smooth or locally constant rationally connected fibration onto a quotient of a compact complex torus.
result Compact Kähler manifolds with pseudo-effective tangent bundles admit a fibration structure.
Local formulas for Poisson structures on wrinkled fibrations are derived.
problem Understanding Poisson structures on specific geometric fibrations.
method Local formulæ for Poisson bivectors and symplectic forms on wrinkled fibrations.
result Local formulas for Poisson structures on wrinkled fibrations are derived.
A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and…
We prove finiteness of hyperkaehler Lagrangian fibrations in any fixed dimension with fixed Fujiki constant and discriminant of the Beauville-Bogomolov-Fujiki lattice, up to deformation. We also prove finiteness of hyperkähler Lagrangian fibrations with an ample line bundle of a given degree on the general fiber of the…
The generic fiber of a Lagrangian fibration on an irreducible holomorphic symplectic manifold is an abelian variety. Associate a polarization type to such Lagrangian fibrations coming from polarizations on a generic fiber. We prove that this polarization type is constant in families of Lagrangian fibrations. Further, w…
The paper connects moment maps to the stability of holomorphic fibrations.
problem Stability of holomorphic fibrations.
method Use of moment maps and K-stability criteria.
result Existence of optimal symplectic connections implies stability of fibrations.
We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants…
Geodesics in Kähler metrics connect metrics with constant scalar curvature.
problem Deriving geodesics for relatively Kähler metrics on fibrations.
method Deriving geodesic equation, proving uniqueness, convexity of log-norm functional.
result Fibrations with optimal symplectic connections are polystable.
Given a Hopf fibration of a round sphere by parallel great subspheres, we prove that the projection map to the base space is, up to isometries of domain and range, the unique Lipschitz constant minimizer in its homotopy class. Similarly, given a Hopf fibration of a round sphere by parallel great circles, we view a unit…
In analogy with the holomorphic case, we compare the topology of Milnor fibrations associated to a meromorphic germ f/g : the local Milnor fibrations given on Milnor tubes over punctured discs around the critical values of f/g, and the Milnor fibration on a sphere.
As groupoids generalize groups, motivated by group extensions we consider a kind of fibrations of Lie groupoids, called locally topological product Lie groupoid fibrations with fiber A, i.e., \[ 1\rightarrow {\sf A} \rightarrow {\sf G} \rightarrow {\sf K}\rightarrow 1 \] where A,G and K are Lie gr…
We construct an explicit diffeomorphism taking any fibration of a sphere by great circles into the Hopf fibration, using elementary geometry--indeed the diffeomorphism is a local (differential) invariant, algebraic in derivatives.
Paper proves existence of weighted constant scalar curvature metrics.
problem Existence of weighted constant scalar curvature Kähler metrics.
method Coercivity of weighted Mabuchi functional implies existence of wcscK metric.
result Equivalence of coercivity and existence of wcscK metrics.
Let p be a finite regular covering on a 2-sphere with at least three branch points. In this paper, we construct a local signature for the class of fibrations whose general fibers are isomorphic to the covering p.
Established a correspondence for toric fibrations using Delzant polytopes.
problem Existence of extremal Kähler metrics on toric fibrations.
method Using weighted constant scalar curvature Kähler metrics and uniform K-stability.
result Equivalence between extremal metrics and weighted uniform K-stability of Delzant polytopes.
In this paper, we consider a special relative Kähler fibration that satisfies a homogenous Monge-Ampère equation, which is called a Monge-Ampère fibration. There exist two canonical types of generalized Weil-Petersson metrics on the base complex manifold of the fibration. For the second generalized Weil-Petersson metri…
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
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, w…
The study of projective varieties with nef anticanonical divisors and log terminal singularities.
problem Understanding the structure and properties of projective varieties with specific divisor conditions.
method Analyzing the Albanese map and MRC fibration for klt projective varieties, showing locally constant fibrations and product decompositions.
result Generalization of results for smooth projective varieties to the klt case, including decomposition into rationally connected and projective varieties with trivial canonical divisor.
Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
problem Characterizing compact locally conformally Kähler manifolds of algebraic codimension one.
method Proving bimeromorphic equivalence to elliptic fibrations.
result Compact LCK manifolds of algebraic codimension one are bimeromorphically equivalent to elliptic fibrations.
New method normalizes Milnor fibrations for real analytic maps.
problem Existence of normalized Milnor fibrations for real analytic maps.
method Introducing a homeomorphism to transform a non-normalized Milnor fibration into a normalized one.
result Normalized map (h−1f)/∣∣h−1f∣∣ defines a smooth locally trivial fibration on the sphere. Quantizes symplectic fibrations to analyze vector bundles and metrics.
problem Quantizing higher rank vector bundles and understanding their metrics.
method Relates Berezin-Toeplitz quantization to hybrid systems and symplectic fibrations.
result Established refined estimates for computing balanced metrics on Kähler manifolds.
It was proved by H. Whitney in 1933 that a Serre fibration of compact metric spaces admits a global section provided every fiber is homeomorphic to the unit interval [0,1]. An extension of the Whitney's theorem to the case when all fibers are homeomorphic to some fixed compact two-dimensional manifold was proved by the…
In the paper we introduce the notions of a singular fibration and a singular Seifert fibration. These notions are natural generalizations of the notion of a locally trivial fibration to the category of stratified pseudomanifolds. For singular foliations defined by such fibrations we prove a de Rham type theorem for the…
Non-archimedean SYZ fibration constructed for Calabi-Yau hypersurfaces.
problem Analyzing Calabi-Yau hypersurfaces using non-archimedean geometry.
method Yamamoto's tropical contractions and Li's Fermat degeneration, with toric plurisubharmonic metrics.
result Constant potential along fibers of retraction under discrete symmetry assumption.
Generalizes tools for studying collapsed manifolds to new geometry.
problem Studying collapsed manifolds with bounded sectional curvature.
method Generalizes fibration and stability theorems for compact group actions on manifolds with local bounded Ricci covering geometry.
result Two generalized results used in Xiaochun Rong's work on almost flat manifolds.
We show that if K is a fibered ribbon knot in S3=∂B4 bounding a ribbon disk D, then given an extra transversality condition the fibration on S3∖ν(K) extends to a fibration of B4∖ν(D). This partially answers a question of Casson and Gordon. In particular, we show the fibration alway…
Locally conformally Hessian manifolds are dense in radiant ones of rank 1.
problem Characterizing locally conformally Hessian manifolds and their properties.
method Analyzing quotient spaces of Hessian manifolds and using statistical manifold theory.
result The set of radiant l.c.H. metrics of rank 1 is dense in all radiant l.c.H. metrics.
Model for associative submanifolds in K3 fibrations.
problem Understanding singularity formation in associative submanifolds.
method Graphs in a 3-manifold with locally gradient flow lines.
result Produces analogues of known singularity formation phenomena.
We explore a number of examples of special Lagrangian fibrations on non-compact Calabi-Yau manifolds invariant under torus actions. These include fibrations on crepant resolutions of canonical toric singularities (already found by Goldstein), proper versions of these fibrations, and fibrations on flat deformations of c…
A smooth fibration of R3 by oriented lines is given by a smooth unit vector field V on R3, 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 ∇V vanishes only in the direction of V.…
Clarifies relation between Pfaffian fibrations and relative algebroids.
problem Understanding geometric structures and symmetries in PDEs.
method Introduces and analyzes Pfaffian fibrations and relative algebroids, clarifying their relationship.
result Every Pfaffian fibration induces a relative algebroid, and their prolongations and local solutions coincide.
In this survey, we remind some fibrations structure theorems (also called Milnor's fibrations) recently proved in the real and complex case, in the local and global settings. We give several Poincaré-Hopf type formulae which relates the Euler-Poincaré characteristic of these fibers (also called Milnor's fibers) and ind…
We study co-associative fibrations of G_{2}-manifolds. We propose that the adiabatic limit of this structure should be given locally by a maximal submanifold in a space of indefinite signature and set up global versions of the constructions.
We establish a product formula for Gromov-Witten invariants for closed, connected, relatively semi-positive Hamiltonian fibrations over any symplectic base. Furthermore, we show that the fibration projection induces a locally trivial (orbi-)fibration map from the moduli space of pseudo-holomorphic maps with marked poin…
Let G⇉M be a source locally trivial proper Lie groupoid such that each orbit is of finite type. The orbit projection M→M/G is a fibration if and only if G⇉M is regular.
There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 loca…
Constructs a new type of metric for elliptic surfaces.
problem Finding a canonical metric on elliptic surfaces.
method Explicit construction of semi-flat constant scalar curvature Kähler current.
result Uniqueness of the semi-flat cscK current under certain conditions.
Extends connections on Lie groupoids, proving completeness conditions.
problem Existence and completeness of multiplicative connections on Lie groupoid fibrations.
method Introduces and investigates multiplicative Ehresmann connections on Lie groupoid fibrations.
result Conditions for completeness of multiplicative connections on Lie groupoid fibrations.
In this paper, we study hyperkahler metric and practice GMN's construction of hyperkahler metric on focus-focus fibrations. We explicitly compute the action-angel coordinates on the local model of focus-focus fibration, and show its semi-global invariant should be harmonic to admit a compatible holomorphic 2-form. Then…
We prove positive mass theorems on ALF manifolds, i.e. complete noncompact manifolds that are asymptotic to a circle fibration over a Euclidean base, with fibers of asymptotically constant length.
We give examples of local signatures, completely different from the usual ones, for general fibrations of genus 2 and genus 3.