arXiv research

Classifies flat projective structures with specific symmetries.

Two projective structures on Riemann surfaces are described and shown not to be identical.

Study on projective structures on a hyperbolic 3-orbifold using tetrahedra.

Characterizes projective special complex manifolds using c-projective structures.

A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifol…

Projective structures are mostly rigid at the boundary but some are not.

3D projective structures can be metrized with conformal structures.

Characterizes monodromies of projective structures on finite-type surfaces.

No projective structure found on foliations of elliptic curves.

Contact projective structures have been profoundly studied by D.J.F. Fox. He associated to a contact projective structure a canonical projective structure on the same manifold. We interpret Fox' construction in terms of the equivalent parabolic (Cartan) geometries, showing that it is an analog of Fefferman's constructi…

A projective structure on a compact Riemann surface X of genus g is given by an atlas with transition functions in PGL(2,C). Equivalently, a projective structure is given by a projective sl(2,C)-bundle over X equipped with a section s and a foliation F which is both transversal to the fibers and the section s. From thi…

Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.

Projective rigidity of circle packings on complex surfaces proved.

We show that the connected sum of two copies of real projective 3-space does not admit a real projective structure. This is the first known example of a connected 3-manifold without a real projective structure.

The paper finds a unique curve minimizing Loewner energy among piecewise geodesic Jordan curves.

We present a number of conditions which are necessary for an n-dimensional projective structure (M,[nabla]) to include the Levi-Civita connection nabla of some metric on M. We provide an algorithm, which effectively checks if a Levi-Civita connection is in the projective class and, in the positive, which finds this con…

Study of affine and projective structures on foliated complex manifolds.

The paper describes projective structures with torsion using formal frames and Thomas-Whitehead connections.

In this article, we introduce symbol calculus on a projective scheme. Using holomorphic Poisson structures, we construct deformations of ring structures for structure sheaves on projective spaces.

We construct an infinite sequence of projectively flat manifolds by using castling transformations of prehomogeneous vector spaces. We also give a classification of manifolds equipped with a flat projective structure obtained by a finite number of castling transformations, and describe these flat projective structures …

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More gene…

By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real p…

Study identifies subvarieties of projective varieties mapping to models.

Symplectic structure found on projective structures on surfaces with boundary.

We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental compon…

3-Sasaki structures linked to projective geometry.

We extend T. Y. Thomas's approach to the projective structures, over the complex analytic category, by involving the $ρ$-connections. This way, a better control of the projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ …

The paper studies conformal and projective structures using 2-frame bundles.

Study real projective structures on a specific Coxeter orbifold.

Bounding geodesic length variation for surface projective structures.

Characterizes representations for complex projective structures with specific branch data.

The paper explores projective structures on curves and their applications in conformal geometry.

Solves open problems on curved projective varieties.

This paper generalizes monodromy maps for projective structures with poles.

We prove that there is a correspondence between projective structures defined by torsion-free connections with skew-symmetric Ricci tensor and Veronese webs on a plane. The correspondence is used to characterise the projective structures in terms of second order ODEs.

Study on projective structures and their foliations on surfaces.

The authors give a complete classification of projective threefolds admitting a holomorphic conformal structure. A Corollary is the complete list of projective threefolds, whose tangent bundle is a symmetric square.

This paper characterizes projective models in statistical relational learning.

Study on projective orbifolds with ends and their deformation theory.

We show that the simultaneous (de)grafting of a complex projective structure with quasi-Fuchsian holonomy along a multicurve can be performed by a simple sequence of one bubbling and one debubbling. As a consequence we obtain that any complex projective structure with quasi-Fuchsian holonomy $ρ:π_1(S)\to$ PSL$_2\mathbb…

We solve the metrisability problem for generic three-dimensional projective structures.

Symplectic coordinates found on projective structures on orbifolds.

This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. Th…

Study on moduli spaces of branched projective structures on surfaces.

The paper classifies certain singular projective varieties with specific properties.

A meromorphic projective structure on a punctured Riemann surface $X\setminus P$ is determined, after fixing a standard projective structure on $X$, by a meromorphic quadratic differential with poles of order three or more at each puncture in $P$. In this article we prove the analogue of Thurston's grafting theorem for…

Study contact structures on projective spaces, proving infinite non-isotopic structures.

The transversal twistor space of a foliation F of an even codimension is the bundle ZF of the complex structures of the fibers of the transversal bundle of F. On ZF, there exists a foliation F' by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I,J) of transversal almost compl…