We formulate the equivalence problem, in the sense of E. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form …
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48 results for “minimal rational tangents”
Study minimal rational curves on complex manifolds with isotropic VMRT.
problem Understanding minimal rational curves tangent to distributions on complex manifolds.
method Partial equivariant compactification of metabelian groups.
result Any isotropic VMRT can be realized as VMRT of minimal rational curves tangent to a distribution.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
Standard embeddings of Schubert varieties in specific manifolds characterized.
problem Characterizing standard embeddings of Schubert varieties in rational homogeneous manifolds.
method Characterization through varieties of minimal rational tangents.
result Characterization of standard embeddings of smooth Schubert varieties.
Complex contact manifolds, varieties of minimal rational tangents, and exterior differential systemsmath.AG
The abstract discusses classification theorems for complex contact manifolds and their geometric properties.
problem Classifying complex contact manifolds and understanding their geometric properties.
method Analyzing contact lines and varieties of minimal rational tangents.
result Partial classification theorems for projective complex contact manifolds.
Study of rational curves in complex manifolds with specific normal bundles.
problem Characterizing rational curves in complex manifolds with given normal bundles.
method Analyzing differential and projective geometric properties of rational curves and their tangents.
result Classification of rational curves into Goursat and Cartan types based on their geometric properties.
The paper classifies surfaces with pseudo-effective tangent bundles.
problem Understanding projective manifolds with pseudo-effective tangent bundles.
method Developed singular hermitian metrics on vector bundles and applied to projective manifolds.
result Projective manifolds with pseudo-effective tangent bundles admit a smooth fibration to a flat projective manifold.
Survey on minimal rational curves and their geometric structures.
problem Germ-equivalence problem of minimal rational curves on uniruled projective manifolds.
method Analysis of isotrivial families of projective varieties and G-structures.
result Natural G-structure on Zariski-open subset of uniruled projective manifolds.
Researchers characterize a specific type of projective variety based on its tangents.
problem Characterizing smooth projective horospherical varieties of Picard number one.
method Using methods of W-normal complete step prolongations and Lie algebra cohomology.
result A uniruled projective manifold of Picard number one is biholomorphic to the variety if its tangents match.
The present paper attempts to show an alternative approach with regards to rational Pythagorean-hodograph (PH) curves and especially more natural approach for rational PH helices (i.e. rational helices). It exploits geometric features of rational helices to obtain a simpler construction of these curves and apply this t…
New construction shows VMRTs of unbendable curves can be Legendrian.
problem Characterize VMRTs of unbendable rational curves under contact structures.
method Used geometry of contact lines and symplectic geometry of distributions.
result VMRTs of Legendrian submanifolds can be realized.
Minimal rational curves on compactified symmetric spaces are orbit-closures of 1-parameter subgroups.
problem When are minimal rational curves on equivariant compactifications of symmetric spaces orbit-closures of 1-parameter subgroups?
method Combining algebraic geometry of minimal rational curves with differential geometry of symmetric spaces, showing Gauss-nondegeneracy of VMRT.
result The Gauss-nondegeneracy of VMRT implies that minimal rational curves on equivariant compactifications of symmetric spaces are orbit-closures of 1-parameter subgroups.
Characterizes symplectic and odd-symplectic Grassmannians using VMRT.
problem Characterizing Fano manifolds of Picard number 1.
method Using VMRT and local differential geometric structure.
result Symplectic and odd-symplectic Grassmannians are characterized by their VMRT.
Study 1-flat G-structures on uniruled projective manifolds.
problem Classify 1-flat irreducible G-structures on uniruled projective manifolds.
method Algebraic geometry and Cartan connections.
result Locally flat structures on VMRTs without 1-flatness assumption.
Paper proves structure for compact Kähler manifolds with pseudo-effective tangent bundles.
problem Compact Kähler manifolds with pseudo-effective tangent bundles.
method Smooth or locally constant rationally connected fibration onto a quotient of a compact complex torus.
result Compact Kähler manifolds with pseudo-effective tangent bundles admit a fibration structure.
The paper extends structure theorem to projective klt varieties with specific tangent sheaf properties.
problem Understanding the structure of projective klt varieties with nef tangent sheaves.
method Developing theory of positivity of coherent sheaves and proving structure theorem.
result Projective klt varieties with specific tangent sheaf properties admit rationally connected fibrations onto abelian varieties.
The paper studies smooth projective varieties with strictly nef tangent bundles and their properties.
problem Understanding smooth projective varieties with strictly nef tangent bundles.
method Analyzing the tangent bundle properties and using rational connectedness.
result Smooth projective varieties with strictly nef tangent bundles are rationally connected.
The paper proves rational connectedness for certain Kähler manifolds.
problem Rational connectedness of compact Kähler manifolds.
method Uniform weak RC-positivity of the tangent bundle.
result Compact Kähler manifolds with uniformly weakly RC-positive tangent bundles are projective and rationally connected.
The paper proves conditions for Kähler and Riemannian manifolds to be simply connected.
problem Conditions for Kähler and Riemannian manifolds to be simply connected.
method Spectral positivity assumptions for Kähler manifolds and a specific spectral positivity assumption for Riemannian manifolds.
result Compact Kähler manifolds and Riemannian manifolds under the specified spectral positivity assumptions are simply connected.
Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Kollár proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces. Our Main Theorem, answering in the affirmative three questions of Kollár, gives s…
Recognizing the -horospherical manifold of Picard number 1 by varieties of minimal rational tangentsmath.AG
A -horospherical manifold is identified by its VMRT.
problem Recognizing -horospherical manifolds of Picard number 1.
method Using the method developed for symplectic Grassmannians, which involves constructing a flat Cartan connection and studying the positivity/negativity of vector bundles.
result The -horospherical manifold ${f X}$ is the only smooth projective variety with the property of being recognized by its VMRT.
The paper explores cone structures and their connections to parabolic geometries in complex manifolds.
problem Understanding cone structures and their properties in complex manifolds.
method Analyzes cone structures induced by parabolic geometries and VMRT structures, focusing on local invariants.
result Establishes a local differential-geometric version of a global algebraic-geometric recognition theorem.
The paper connects curvature positivity to rational connectedness in complex geometry.
problem Establishing a geometric criterion for rational connectedness.
method Uhlenbeck-Yau's continuity method applied to mean curvature positivity.
result Holomorphic tangent bundle mean curvature positivity is equivalent to rational connectedness of compact Kähler manifolds.
Rigidity of irreducible Hermitian symmetric spaces of the compact type under K"ahler deformationmath.AG
We study deformations of irreducible Hermitian symmetric spaces of the compact type, known to be locally rigid, as projective-algberaic manifolds and prove that no jump of complex structures can occur. For each of rank there is an associated reductive linear group such that admits a holomorphic …
Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers o…
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…
A valuation minimizes volume for K-semistable singularities.
problem Stability of valuations in higher rational rank singularities.
method Analyzing quasi-monomial valuations and their associated graded rings.
result A minimizer of the normalized volume function is unique and corresponds to K-semistable singularities.
A method is proposed to construct spiral curves by inversion of a spiral arc of parabola. The resulting curve is rational of 4-th order. Proper selection of the parabolic arc and parameters of inversion allows to match a wide range of boundary conditions, namely, tangents and curvatures at the endpoints, including thos…
Vanishing theorems show holomorphic tensor fields on certain Kähler manifolds are trivial.
problem Understanding properties of holomorphic tensor fields on Kähler manifolds.
method Established vanishing theorems for uniformly rational connected (RC) -positive Hermitian holomorphic vector bundles.
result Holomorphic tangent bundles of Kähler manifolds with positive -Ricci curvature are uniformly RC -positive.
Solves open problems on curved projective varieties.
problem Structure theorems for curved projective varieties.
method Supplements and proposes open problems.
result Provides new insights into structure of curved projective varieties.
Researchers prove a complex geometric conjecture about certain manifolds.
problem Compact simply connected Riemannian manifolds with nonnegative sectional curvature.
method Assumption of entire Grauert tube and real analytic structure.
result Compact simply connected Riemannian manifolds with entire Grauert tube are rationally elliptic.
Study shows how neck pinches occur in Lagrangian flows and their continuation.
problem Understanding and continuation of Lagrangian mean curvature flows with singularities.
method Analyzes zero Maslov, rational Lagrangian flows in compact Calabi-Yau surfaces.
result Tangent flow is unique and can be continued past singularities.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tange…
Study of holomorphic distributions on projective 3-space, focusing on stable tangent sheaves.
problem Characterizing holomorphic distributions and their tangent sheaves on projective 3-space.
method Analysis of singular schemes and tangent sheaves, classification of distributions, use of Grothendieck's Quot-scheme.
result Classification of codimension one distributions with stable tangent sheaves and description of moduli spaces.
Formula conjectured for rational cuspidal curves in projective plane.
problem Counting rational cuspidal curves in projective plane.
method Extending Kontsevich's recursion formula and using geometric input about tangency of curves at nodal points.
result Conjectural formula agrees with earlier computations and extends to rational quartics with E6 singularity.
The paper reconfirms a lower bound on rational genus using Heegaard Floer homology.
problem Lowering the rational genus of knots in rational homology 3-spheres.
method Using Heegaard Floer homology and the -invariant.
result Same lower bound and minimizers as Ni and the first author's results.
Paper classifies rational 3-tangles using normal forms and minimal coordinates.
problem Classifying rational 3-tangles up to isotopy.
method Defined normal form and normal coordinate, investigated minimal coordinates, constructed contractible simplicial complex.
result Simplicial complex of normal forms is contractible, leading to classification of rational 3-tangles.
Unique tangent cones found for boundary points of 2D almost-minimizing currents.
problem Characterizing boundary points of two-dimensional almost-minimizing currents.
method Combining epiperimetric inequality and almost-monotonicity formula.
result Tangent cones at singular boundary points are unique.
Minimal hypersurfaces with cylindrical tangent cones constructed and analyzed.
problem Constructing minimal hypersurfaces with specific geometric properties.
method Constructing minimal hypersurfaces with cylindrical tangent cones and proving unique continuation results.
result Existence and properties of minimal hypersurfaces with cylindrical tangent cones.
The paper proves -convergence of discrete tangent-point energies to continuous energies and ropelength, with applications to biarc curves.
problem Proving convergence of discrete tangent-point energies to continuous energies and ropelength.
method Using biarc curves and interpolation, the paper proves -convergence of discretized tangent-point energies to the continuous tangent-point energies and ropelength functional.
result Discrete almost minimizing biarc curves converge to ropelength minimizers and minimizers of continuous tangent-point energies.
We study singularities of Lagrangian mean curvature flow in $\C^n$ when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonia…
The paper studies Kähler manifolds with partially semi-positive curvature and rational connectedness.
problem Analyzing compact Kähler manifolds with partially semi-positive curvature and rational connectedness.
method Proving rational connectedness for manifolds with BC- positive tangent bundles, and applying these results to curvature conditions.
result Confirming a conjecture and generalizing results on rational connectedness and curvature conditions.
Study shows special Kähler geometry on base of holomorphic Lagrangian fibrations implies projective space.
problem Understanding the geometry of holomorphic Lagrangian fibrations.
method Using special Kähler geometry and parallel splitting of tangent bundles.
result Holomorphic Lagrangian fibrations over non-projective spaces are impossible.
Simplified presentation of symplectic fillings of lens spaces.
problem Symplectic fillings of lens spaces.
method Visual presentation of rational blowdown algorithm using triangulations of convex polygons.
result Organization of symplectic fillings into a graph.
Uniqueness proven for cylindrical tangent cones in high dimensions.
problem Proving uniqueness of cylindrical tangent cones in high dimensions.
method Analyzing area-minimizing hypersurfaces in R^9.
result Uniqueness of cylindrical tangent cones in .
Minimal surfaces' behavior at infinity resolved.
problem Asymptotic behavior of minimal surfaces with quadratic area growth.
method Partial resolution of a conjecture by Meeks.
result Criterion for uniqueness of tangent cones at infinity.
Study proves uniqueness of tangent cones for area-minimizing currents in higher codimensions.
problem Understanding the fine structure of singular points in area-minimizing currents.
method Analysis of tangent cones and application of previous work.
result Uniqueness of tangent cones at -a.e. points in the support of area-minimizing currents.
Study shows unique tangent cones for area-minimizing currents at boundary points.
problem Uniqueness of tangent cones for area-minimizing currents with arbitrary multiplicity.
method Analysis of area minimizing currents in submanifolds with arbitrary boundary multiplicity.
result Tangent cones are unique at density boundary points.