arXiv research

The paper classifies fibrations of 3-dimensional flat orbifolds.

This paper classifies fibrations of flat orbifolds, advancing flat 4-manifold classification.

Calculates characteristic classes of flat vector bundles on complex fibers.

In this paper, we consider an equivalence problem of second order partially differential equations (PDE) and a duality of the flat differential equation. For the equivalence problem, explicit form of invariants (curvatures) are given. We also investigate a duality associated with the flat equation using double fibratio…

The paper studies curvature properties of Monge-Ampère fibrations and their existence.

Flat tori in 3-sphere have π extrinsic diameter under specific conditions.

Study Higgs sections and flat sections for nonlinear harmonic bundles.

Formula connects analytic torsion forms of fibration and its pieces.

The paper studies the geometry of flat manifolds with controlled holonomy.

In this paper we describe the classification of all the geometric fibrations of a closed flat Riemannian 4-manifold over a 1-orbifold.

We investigate complete noncompact Ricci-flat manifolds which are not of maximal volume growth. We show that the manifolds with a curvature decay condition and a holonomy decay condition are asymptotic to torus fibrations over ALE spaces. In particular, we classify complete noncompact 4-dimensional hyperkäler manifold…

Constructs a new type of metric for elliptic surfaces.

We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampere equation, we show that the Ricci-flat metrics collapse (away f…

Generalizes tools for studying collapsed manifolds to new geometry.

We show that under some non-degeneracy assumption the only submersive harmonic morphism on a conformally flat $3-$sphere is the Hopf fibration. The proof involves an appropriate use the Chern-Simons functional.

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…

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…

Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi construction, that is, E^n/G is represented as the quotient of the Cartesian product …

A fibration of ${\mathbb R}^n$ by oriented copies of ${\mathbb R}^p$ is called skew if no two fibers intersect nor contain parallel directions. Conditions on $p$ and $n$ for the existence of such a fibration were given by Ovsienko and Tabachnikov. A classification of smooth fibrations of ${\mathbb R}^3$ by skew oriente…

Study Cayley fibrations on Bryant-Salamon $Spin(7)$ manifolds.

Asymptotically flat manifolds with Euclidean volume growth are known to be ALE. In this paper, we consider a class of asymptotically flat manifolds with slower volume growth and prove that their asymptotic geometry is that of a fibration over an ALE manifold. In particular, we show that gravitational instantons with cu…

Given the Lagrangian fibration $T^4\to T^2$ and a Lagrangian submanifold, exhibiting an elliptic umbilic and supporting a flat line bundle, we study, in the context of mirror symmetry, the ``quantum'' corrections necessary to solve the monodromy of the holomorphic structure of the mirror bundle on the dual fibration.

Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.

We present explicit constructions of complete Ricci-flat Kahler metrics that are asymptotic to cones over non-regular Sasaki-Einstein manifolds. The metrics are constructed from a complete Kahler-Einstein manifold (V,g_V) of positive Ricci curvature and admit a Hamiltonian two-form of order two. We obtain Ricci-flat Ka…

We study the natural structure on the moduli space of deformations of compact coassociative submanifolds. We show that a G2-manifold with a T^4-action of isomorphisms such that the orbits are coassociative tori is locally equivalent to a minimal 3-manifold in R^{3,3} = H^2(T^4,R) with positive induced metric. By studyi…

Study singular fibers in genus 2 algebraic fibrations and their monodromy factorizations.

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the re…

The paper connects Kähler-Ricci shrinkers to Fano fibrations in algebraic geometry.

In this article we use the adiabatic 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 associated to the fibration of the cutting hypersurface are vanished. In this paper we assume that the m…

Study finite group actions on Riemann surfaces to understand branes in Higgs bundles.

The paper finds new Ricci solitons from Hopf fibrations.

Explains Higgs bundles and their applications in gauge theory.

Motivated by the picture of mirror symmetry suggested by Strominger, Yau and Zaslow, we made a conjecture concerning the Gromov-Hausdorff limits of Calabi-Yau n-folds (with Ricci-flat Kähler metric) as one approaches a large complex structure limit point in moduli; a similar conjecture was made independently by Kontsev…

A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove…

A new liquid crystalline texture is proposed using gnomonic projection of the Hopf fibration.

The paper studies foliations on smooth projective varieties and their properties.

Two remarks on curvature properties of Kähler manifolds.

For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the…

We describe pairs (p,n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms.

Characterizes curvature positivity for Riemannian metrics on flat vector bundles.

One of the main results of the paper arXiv:1108.0967 by Gross-Tosatti-Zhang establishes estimates on the collapsing of Ricci-flat Kahler metrics on holomorphic torus fibrations. We remove a projectivity assumption from these estimates and simplify some of the underlying analysis.

Proves SYZ mirror symmetry for del Pezzo and rational elliptic surfaces.

The paper proves a section for Anosov vector fields on compact manifolds.

We study the behaviour of analytic torsion under smooth fibrations. Namely, let F \to E \to^{f} B be a smooth fiber bundle of connected closed oriented smooth manifolds and let $V$ be a flat vector bundle over $E$. Assume that $E$ and $B$ come with Riemannian metrics and $V$ comes with a unimodular (not necessarily fla…

The abstract proves a conjecture about geometric structures in Calabi-Yau orbifolds.

Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.

Log-conformal projective pairs restrict to simple geometric structures.

The paper classifies surfaces with pseudo-effective tangent bundles.