arXiv research

Formula derived for a magnetic line invariant.

Invariant Kähler metrics on line bundles are derived from the Calabi ansatz.

We recall the definition of the quadratic helicity invariant and of the higher asymptotic ergodic $M$-invariant. We present a simpler new proof (in part) that the $M$-invariant is ergodic. The $M$-invariant is a higher invariant, this means that for the magnetic field with closed magnetic lines the invariant is not a f…

Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements a…

Study of line bundles on spherical varieties leads to Calabi-Yau metrics.

Only products of projective lines have vanishing Futaki invariants for all Kähler classes.

We generalise the Kreck-Stolz invariants s_2 and s_3 by defining a new invariant, the t-invariant, for quaternionic line bundles E over closed spin-manifolds M of dimension 4k-1 with H^3(M; \Q) = 0 such that c_2(E)\in H^4(M) is torsion. The t-invariant classifies closed smooth oriented 2-connected rational homology 7-s…

We explore the connection between Hilbertian metrics and positive definite kernels on the real line. In particular, we look at a well-known characterization of translation invariant Hilbertian metrics on the real line by von Neumann and Schoenberg (1941). Using this result we are able to give an alternate proof of Boch…

The study defines invariants for time-like surfaces with real asymptotic lines.

Paper proves certain closed affine manifolds without invariant lines don't exist.

Study Miyaoka-Yau inequality for certain projective manifolds.

We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by const…

We study eta-invariants on odd dimensional manifolds with boundary. The dependence on boundary conditions is best summarized by viewing the (exponentiated) eta-invariant as an element of the (inverse) determinant line of the boundary. We prove a gluing law and a variation formula for this invariant. This yields a new, …

Global theory of relative invariants and equivariant line bundles established.

We first describe the numerical invariants attached to the second fundamental form of a spacelike surface in four-dimensional Minkowski space. We then study the configuration of the nu-principal curvature lines on a spacelike surface, when the normal field nu is lightlike (the lightcone configuration). Some observation…

New invariant detects non-homeomorphic arrangements with similar coefficients.

New invariant identifies complex line arrangements with same combinatorics but different embeddings.

We prove a conjecture of Crapo and Penne which characterizes isotopy classes of skew configurations with spindle-structure. We use this result in order to define an invariant, spindle-genus, for spindle-configurations. We also slightly simplify the exposition of some known invariants for configurations of skew lines an…

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both $L^2$ combinatorial and $L^2$ analytic torsion invariants …

We define a new topological invariant of line arrangements in the complex projective plane. This invariant is a root of unity defined under some combinatorial restrictions for arrangements endowed with some special torsion character on the fundamental group of their complements. It is derived from the peripheral struct…

Geometric Invariant Theory applied to Kähler manifolds yields analytic models for vector bundles.

New invariant distinguishes tight contact structures on 3-tori.

On a compact Kahler manifold, one can define global invariants by integrating local invariants of the metric. Assume that a global invariant thus obtained depends only on the Kahler class. Then we show that the integrand can be decomposed into a Chern polynomial (the integrand of a Chern number) and divergences of one …

Introduces a new CR invariant for co-oriented contact structures on closed 3-manifolds

We provide a proof and analyze the asymptotic behavior of a formula for the linking number of line segments.

Motivated by the work of Abreu and Freitas, we study the invariant spectrum of the Laplace operator associated to hermitian line bundles endowed with invariant metrics over $\mathbb{p}^1$.

We study differential invariants of linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles over smooth manifolds with respect to groups of authomorphisms.

A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a topological invariant of complexified real line arrangements, the chamber weight. This in…

Recently L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (whose link $M$ is a rational homology sphere) with the Seiberg-Witten invariant of $M$ associated with the ``canonical'' $spin^c$ structure of $M$. Since the Seiberg-Witten t…

This article gives matrix factorizations for the trivalent diagrams and double line appearing in $\mathfrak{sl}_n$ quantum link invariant. These matrix factorizations reconstruct Khovanov-Rozansky homology. And we show that the Euler characteristic of the matrix factorization for a double loop equals the quantum dimens…

We give quantitative and qualitative results on the family of surfaces in $\mathbb{CP}^3$ containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines $E$. We prove that its general element is a smooth surface containing $E$ and no other line. Afterwards we prove that …

New Einstein metrics constructed on complex line bundle over CP1.

The study classifies holomorphic projective connections on complex threefolds.

We associate a Taylor tower supplied by calculus of the embedding functor to the space of long knots and study its cohomology spectral sequence. The combinatorics of the spectral sequence along the line of total degree zero leads to chord diagrams with relations as in finite type knot theory. We show that the spectral …

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computin…

We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our r…

In this paper we study the behavior of the Ricci flow at infinity for the full flag manifold $SU(3)/T$ using techniques of the qualitative theory of differential equations, in special the Poincaré Compactification and Lyapunov exponents. We prove that there are four invariant lines for the Ricci flow equation, each one…

The study constructs universal invariants for non-Archimedean metrics on projective varieties.

Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant $t^{I\left( \mathcal{L} \right) }$ is constructed for a link $\mathcal{L}$, where $I$ is the abelian Chern-Simons action and $t$ a formal constant. For oriented knotted vortex lines, $t^{I}$ satisf…

We study the second order invariants of a Lorentzian surface in $\mathbb{R}^{2,2},$ and the curvature hyperbolas associated to its second fundamental form. Besides the four natural invariants, new invariants appear in some degenerate situations. We then introduce the Gauss map of a Lorentzian surface and give an extrin…

Real del Pezzo surfaces split real lines into elliptic and hyperbolic types.

Geometric quantization extended to big line bundles.

As in the case of minimal surfaces in the Euclidean 3-space, the reflection principle for maximal surfaces in the Lorentz-Minkowski 3-space asserts that if a maximal surface has a spacelike line segment $L$, the surface is invariant under the $180^\circ$-rotation with respect to $L$. However, such a reflection property…

Minimal surfaces in Heisenberg group have null curves and lines.

The study examines conditions that prevent null geodesic lines in spacetimes, impacting cosmological geometry.

Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.

In this paper we prove that a properly embedded constant mean curvature surface in $\mathbb{H}^2\times\mathbb{R}$ which has finite topology and stays at a finite distance from a vertical geodesic line is invariant by rotation around a vertical geodesic line.

We study real lines on certain Moishezon threefolds which are potentially twistor spaces of 3CP^2. Here, line means a smooth rational curve whose normal bundle is O(1)^2 and the reality implies the invariance under an anti-holomorphic involution on the threefolds. Our threefolds are birational to double coverings of CP…