arXiv research

Computes the decomposition of rank-three bundles over the projective line with three marked points.

The paper studies how points and lines can move while preserving incidences.

The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …

We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.

Study smooth embeddings of line configurations in complex projective plane.

Study projective flat vector bundles over Riemann surfaces using Wronskian line bundles.

Study Miyaoka-Yau inequality for certain projective manifolds.

Weil-Petersson volumes vary continuously with weighted points on a projective line.

Extended Einstein manifolds reveal new symmetries.

In this note we prove a decomposition related to the affine fundamental group and the projective fundamental group of a line arrangement and a reducible curve with a line component. We give some applications to this result.

We show that a generic real projective n-dimensional hypersurface of degree 2n-1 contains "many" real lines, namely, not less than (2n-1)!!, which is approximately the square root of the number of complex lines. This estimate is based on the interpretation of a suitable signed count of the lines as the Euler number of …

We prove that the Fourier--Laplace--Nahm transform for connections on the projective line is a hyper-Kähler isometry.

The study classifies semiaffine stable planes into affine, projective, or punctured projective planes.

The study classifies holomorphic projective connections on complex threefolds.

Establishes metrics with positive curvature on projective line bundles.

Classifies and constructs intertwining differential operators between line and vector bundles over real projective space.

The paper classifies and constructs differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces.

Characterizes projective submanifolds in high dimensions.

Study weak geodesic lines in Kähler metric space, disproving a conjecture.

Study shows Ricci flat Calabi's metrics can't be projectively induced.

Study on Kähler manifolds with non-positive mixed curvature and its implications.

Study of line congruences for Appell's rank-4 hypergeometric functions.

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

This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.

Study on positivity of CM line bundles on moduli space of klt good minimal models with κ=1.

Let X be a complex-projective contact manifold whose second Betti-number is one. It has long been conjectured that X should then be rational-homogeneous, or equivalently, that there exists an embedding of X into a projective space whose image contains lines. Using methods introduced in math.AG/0206193, we show that X i…

The "dancing metric" is a pseudo-riemannian metric $\pmb{g}$ of signature $(2,2)$ on the space $M^4$ of non-incident point-line pairs in the real projective plane $\mathbb{RP}^2$. The null-curves of $(M^4,\pmb{g})$ are given by the "dancing condition": the point is moving towards a point on the line, about which the li…

We propose a geometric correspondence between (a) linearly degenerate systems of conservation laws with rectilinear rarefaction curves and (b) congruences of lines in projective space whose developable surfaces are planar pencils of lines. We prove that in projective 4-space such congruences are necessarily linear. Bas…

Study on the asymptotic geometry of Higgs bundles over projective line.

Extends complex manifold structures to line bundles, revealing new projective manifolds.

We show that the Debarre-de Jong conjecture that the Fano scheme of lines on a smooth hypersurface of degree at most n in n-dimensional projective space must have its expected dimension, and the Beheshti-Starr conjecture that bounds the dimension of the Fano scheme of lines for hypersurfaces of degree at least n in n-d…

We prove that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines which have disconnected realization spaces.

We give an explicit description of rational curves in the product of three copies of complex projective lines, which are transformed into twistor lines in M. Nagata's example of non-projective complete algebraic variety, viewed as the twistor space of Eguchi-Hanson metric. In particular, we show that there exist two fa…

A new snake model improves segmentation of SEM images.

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

The paper characterizes and studies compact subsets of complex projective space with specific line intersection properties.

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology of the divisor complement determined by its mixed Hodge structure. We then solve …

In this paper are given examples of tori T2 embedded in R3 with all their principal lines dense. These examples are obtained by stereographic projection of deformations of the Clifford torus in S3.

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…

Let $Γ$ be a surface group of higher genus. Let $ρ\_0: Γ\to {PGL}(V)$ be a discrete faithful representation with image contained in the natural embedding of ${SL}(2, {\mathbb R})$ in ${PGL}(3, {\mathbb R})$ as a group preserving a point and a disjoint projective line in the projective plane. We prove that such a repres…

A venerable problem in combinatorics and geometry asks whether a given incidence relation may be realized by a configuration of points and lines. The classic version of this would ask for algebraic lines over some field or possibly real pseudolines: embedded circles (isotopic to $RP^1$) in the real projective plane. In…

Study on constraints for topological and smooth realizations of line arrangements and configurations.

Study foliations in PSL(4,R)-Teichmüller theory, proving two invariant foliations.

The paper studies scalar flat Kähler metrics on line bundles and proves their properties.

Constructs a topological cover of real line's multiplicative group.

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on the complex projective plane with singularities at a colle…

We study the geometry of the cuspidal edge $M$ in $\mathbb R^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the cont…

We study holomorphic locally homogeneous geometric structures modelled on line bundles over the projective line. We classify these structures on primary Hopf surfaces. We write out the developing map and holonomy morphism of each of these structures explicitly on each primary Hopf surface.