arXiv research

Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.

Paper proves achiral Lefschetz fibrations from non-orientable Lefschetz fibrations.

We develop some theory of double fibration transforms where the cycle space is a smooth manifold and apply it to complex projective space.

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…

Study normal operators of double fibration transforms with conjugate points.

Study groups of order 64 and non-homeomorphic double Kodaira fibrations with same invariants.

We establish a duality within the spectral sequence that governs the holomorphic double fibration transform. It has immediate application to the questions of injectivity and range characterization for this transform. We discuss some key examples and an improved duality that holds in the Hermitian holomorphic case.

New groups from surface braids help create complex geometric shapes.

New singularities and fibrations in non-orientable 4-manifolds.

Develops support theorem for analytic transforms in tomography.

We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus $g\geq 3$ and extend these involutions to the four-manifolds obtained by blowing up the …

We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $Σ_b \times Σ_b$, where $Σ_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism from the pure braid group $\mathsf{P}_2(Σ_b)$ to some finite Heisenberg group.…

Transforms uniquely determine Higgs fields on real-analytic manifolds.

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fibre bundles. They are known to have positive signature divisible by $4$. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent…

A global twistor correspondence is established for neutral self-dual conformal structures with alpha-surface foliation when the structure is close to the standard structure on S^2 times S^2. We need to introduce some singularity for the alpha-surface foliation such that the leaves intersect on a fixed two sphere. In th…

Study X-ray transform on manifolds, desingularize, and improve mapping properties.

The study constructs and analyzes new symplectic 4-manifolds from hyperelliptic Lefschetz fibrations.

A class of examples of Riemannian metrics with holonomy G_2 on compact 7-manifolds was constructed by the author in arXiv:math.DG/0012189 and later in a joint work with N.-H. Lee in arXiv:0810.0957, using a certain `generalized connected sum' of two asymptotically cylindrical manifolds with holonomy SU(3). We consider,…

The Hopf fibration mapping circles on a 3-sphere to points on a 2-sphere is well known to topologists. While the 2-sphere is embedded in 3-space, four-dimensional Euclidean space is needed to visualize the 3-sphere. Visualizing objects in 4-space using computer graphics based on their analytical representations has bec…

The paper explores conditions for sections in Lefschetz fibrations and bundles over 2-complexes.

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz fibrations over the 2-sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1-hand…

This is the final part of the work started in math.DG/0611281 and math.DG/0703916. Here the question of double fibration ois adressed both for relative k-theory and free multiplicative K-theory. In the case of relative and ``nonfree'' multiplicative K-theory, the direct image is proved to be functorial for double subme…

We describe the $ν$-lines of curvature of an embedding of the double torus into $\mathbb R^4$, defined as the link of the real part of the Milnor fibration of a polynomial, where $ν$ is its gradient. Through this analysis, we present a complete description of the foliation of lines of curvature of the embedding, define…

The paper connects sectional category and parametrized Borsuk-Ulam property for fibrations.

Study on determining metrics from minimal surface areas, extending earlier work.

New method learns operators with geometric singularities from few samples.

The basic setup consists of a complex flag manifold $Z=G/Q$ where $G$ is a complex semisimple Lie group and $Q$ is a parabolic subgroup, an open orbit $D = G_0(z) \subset Z$ where $G_0$ is a real form of $G$, and a $G_0$--homogeneous holomorphic vector bundle $\mathbb E \to D$. The topic here is the double fibration tr…

This paper constructs explicit trisection diagrams for elliptic surfaces.

We present an approach to constructing broken Lefschetz fibrations (BLFs) $f:X\rightarrow S^2$ from a handle decomposition of a 4-manifold $X$. Given a handle decomposition as input these techniques yield explicit descriptions of the BLFs, and do not rely on classification results from contact topology or the choice of…

The article studies mapping properties of Radon transform and backprojection on a unit ball.

New exotic 4-manifolds with even $b_2^+$ and $\mathbb{Z}/2\mathbb{Z}$ fundamental group.

Nonorientable 4-manifolds can be fibred over 2-disks with nonorientable fibers.

For sutured 3-manifolds M, there is a sutured Thurston norm due to Scharlemann. We show how depth one foliations of M and corresponding fibrations and the usual Thurston norm on the double of M are useful tools for computing this norm. In many examples, the faces of the unit ball of the sutured norm are related to cone…

In this paper we obtain the following results: (1) Any compact Stein surface with boundary embeds naturally into a symplectic Lefschetz fibration over the 2-sphere. (2) There exists a minimal elliptic fibration over the 2-disk, which is not Stein. (3) The circle bundle over a genus n>1 surface with euler number e=-1 ad…

This paper investigates some actions "à la Johnson" on the set, denoted by ${\cal E}$, of Spin-structures which are interpreted as special double-coverings of a trivial $S^1-$fibration over a non-orientable surface $N_{g+1}$. The group acting is first a group of orthogonal isomorphisms assoiciated to $N_{g+1}$. A secon…

We construct exact sequences of invariant differential operators acting on sections of certain homogeneous vector bundles in singular infinitesimal character, over the isotropic $2$-Grassmannian. This space is equal to $G/P$, where $G$ is $\operatorname{Sp}(2n,\mathbb{C})$, and $P$ its standard parabolic subgroup havin…

We calculate the eta-invariant for the odd signature operator relative to a specific submersion metric on the Milnor fibration of a quasihomogeneous hypersurface singularity using certain global boundary conditions in terms of the data of its fibre intersection form, monodromy and variation mapping resp. the monomial d…

Hyperkahler manifolds with round Kahler cones have unique bimeromorphic models.

Study nondegenerate fibrations of Euclidean spaces and their relation to sphere fibrations.

We investigate the structure of a variety of new Moishezon twistor spaces, by utilizing the pluri-half-anti-canonical map from the twistor spaces. Each of these twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of…

Introduces new stability concept for Fano fibrations.

Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.

This thesis explores Hamiltonian systems and Kähler structures on complex coadjoint orbits.

Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism type…

A fibration of $\mathbb{R}^3$ by oriented lines is given by a unit vector field $V : \mathbb{R}^3 \to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationship…

A simplified trisection is a trisection map on a 4-manifold such that, in its critical value set, there is no double point and cusps only appear in triples on innermost fold circles. We give a necessary and sufficient condition for a 3-tuple of systems of simple closed curves in a surface to be a diagram of a simplifie…

Same genus-2 fibration structures for specific types found by different researchers.