arXiv research

Study of line fibrations in R^3, focusing on non-skew cases.

Standard contact structure proven for line fibration in 3D space.

Contact structures are induced by nondegenerate skew fibrations of R^3.

Spin Lefschetz fibrations can represent any group and lattice point.

The study describes Lefschetz-Bott fibrations on symplectic manifolds and their applications.

Model for associative submanifolds in K3 fibrations.

Describes curvature lines on a double torus in 4D space.

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…

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…

Reciprocity laws for line bundles on circle fibrations over complex manifolds.

Stable Higgs bundles on elliptic fibrations are decomposed into simpler components.

The paper studies geodesic-Einstein metrics on line bundles over fibration.

Study Fano fibrations and Kähler-Einstein metrics on their bases.

The study calculates the index distribution of Brownian loops in various geometrical settings.

Shows CM line bundles are ample on K-stable varieties.

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

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

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

Constructs new coassociative fibrations for G2 manifolds.

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.

In their previous work, Barraud and Cornea enriched the Lagrangian Floer complex by adding cubical chains in the based loop space of the Lagrangian, and recovered the Leray-Serre spectral sequence of the based path space fibration, assuming that the Lagrangian is weakly exact and simply connected. In the present articl…

Researchers reinterpret complex hyperbolic orbifolds using line arrangements.

In this paper we study the Milnor fibrations associated to real analytic map germs $ψ:(\mathbb{R}^{m},0) \to (\mathbb{R}^2,0)$ with isolated critical point at $0\in \mathbb{R}^{m}$. The main result relates the existence of called Strong Milnor fibrations with a transversality condition of a convenient family of analyti…

Study on Kähler-Ricci flow's infinite-time singularities.

The paper simplifies proofs and characterizes contact structures in 3D.

Study shows scalar curvature of a specific type of manifold converges to -m outside singular points.

Researchers create a projective space for quasimaps and study its connections to Calabi-Yau fibrations.

The purpose of this paper is first to give an asymptotic formula for the holomorphic analytic torsion forms of a fibration associated with increasing powers of a given line bundle. Secondly, we generalize this formula, thanks to the theory of Toeplitz operators, in the case where the powers of the line bundle is replac…

Paper proves direct image sheaf positivity for certain Kähler fibrations.

In this paper, we study rational sections of the relative Picard scheme of a linear system on a smooth projective variety. We prove that if the linear system is basepoint-free and the locus of non-integral divisors has codimension at least two, then all rational sections of the relative Picard scheme come from restrict…

We list all analytic diffeomorphisms between an open subset of the 4-dimensional projective space and an open subset of the 4-dimensional sphere that take all line segments to arcs of round circles. These are the following: restrictions of the quaternionic Hopf fibrations and projections from a hyperplane to a sphere f…

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

Established a correspondence for toric fibrations using Delzant polytopes.

In this paper we shall prove that the plane of financial events, introduced and applied to financial problems by the author himself (see [2], [3] and [4]) can be considered as a fibration in two different ways. The first one, the natural one, reveals itself to be isomorphic to the tangent- bundle of the real line, when…

Let $M$ be a hyperkaehler manifold, and $η$ a closed, positive (1,1)-form which is degenerate everywhere on $M$. We associate to $η$ a family of complex structures on $M$, called a degenerate twistor family, and parametrized by a complex line. When $η$ is a pullback of a Kaehler form under a Lagrangian fibration $L$, a…

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is gene…

Study monodromy factorizations for lines on del Pezzo surfaces.

Study the monodromy and center-focus problems for rational maps defined by products of generic lines.

For smooth families of projective algebraic curves, we extend the notion of intersection pairing of metrized line bundles to a pairing on line bundles with flat relative connections. In this setting, we prove the existence of a canonical and functorial "intersection" connection on the Deligne pairing. A relationship is…

In this paper, we study the geometry of the SYZ transform on a semi-flat Lagrangian torus fibration. Our starting point is an investigation on the relation between Lagrangian surgery of a pair of straight lines in a symplectic 2-torus and extension of holomorphic vector bundles over the mirror elliptic curve, via the S…

Let X --> B be a holomorphic submersion between compact Kahler manifolds of any dimension, whose fibres and base have no non-zero holomorphic vector fields and whose fibres all admit constant scalar curvature Kahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar cu…

In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Dona…

Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.

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 give an explicit construction of complex maps whose nodal line have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, $\ell$) Lissajous figure, and are therefore a subfamily of spiral knots generalising the torus…

The Seiberg-Witten family of elliptic curves defines a Jacobian rational elliptic surface $\Z$ over $\mathbb{C}\mathrm{P}^1$. We show that for the $\bar{\partial}$-operator along the fiber the logarithm of the regularized determinant $-1/2 \log \det' (\bar\partial^* \bar\partial)$ satisfies the anomaly equation of the …

Let $π:\mathcal{X}\to M$ be a holomorphic fibration with compact fibers and $L$ a relatively ample line bundle over $\mathcal{X}$. We obtain the asymptotic of the curvature of $L^2$-metric and Qullien metric on the direct image bundle $π_*(L^k\otimes K_{\mathcal{X}/M})$ up to the lower order terms than $k^{n-1}$ for la…

Study geometric quantization and measured Gromov-Hausdorff convergence of frame bundle metrics.