arXiv research

Smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 are horospherical varieties. We characterize standard embeddings of smooth Schubert varieties in rational homogeneous manifolds of Picard number 1 by means of varieties of minimal rational tangents. In particular, we mainly consider nonhomog…

Smooth Schubert varieties are rigid in rational homogeneous manifolds.

Strong formal properties for toric and homogeneous Kähler manifolds.

New proof for curved 3-cohom manifold rational ellipticity.

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…

The paper characterizes symplectic fillings of Seifert 3-manifolds using rational blowdowns.

The study classifies manifolds with specific rational cohomology properties.

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to …

Study splitting submanifolds in specific homogeneous spaces.

New Stein fillings found for non-weighted homogeneous singularities.

The paper solves the dHYM equation on rational homogeneous varieties using Lie theory.

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…

The paper describes spectra of operators on rational homogeneous varieties.

We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…

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…

Study on metrics of non-negative curvature on vector bundles over specific manifolds.

A ${ m G}_2$-horospherical manifold is identified by its VMRT.

In this paper we prove the following result: if two 2-dimensional 2-homogeneous rational vector fields commute, then either both vector fields can be explicitly integrated to produce rational flows with orbits being lines through the origin, or both flows can be explicitly integrated in terms of algebraic functions. In…

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…

We give new counterexamples to a question of Karsten Grove, whether there are only finitely many rational homotopy types among simply connected manifolds satisfying the assumptions of Gromov's Betti number theorem. Our counterexamples are homogeneous Riemannian manifolds, in contrast to previous ones. They consist of t…

New examples show deletion type admissible pairs can be rigid under rational saturation.

The paper classifies minimal symplectic fillings of small Seifert 3-manifolds.

The Ricci flow is a parabolic evolution equation in the space of Riemannian metrics of a smooth manifold. To some extent, Einstein equations give rise to a similar hyperbolic evolution. The present text is an introductory exposition to Bianchi-Ricci and Bianchi-Einstein flows, that is, the restricted finitely dimension…

We introduce a new class of zero-dimensional weighted complete intersections, by abstracting the essential features of rational cohomology algebras of equal rank homogeneous spaces of compact connected Lie groups. We prove that, on a 1-connected closed manifold M whose rational cohomology algebra belongs to this class,…

Study Kähler-Ricci flow on rational homogeneous varieties using algebraic geometry and representation theory.

Study shows no hyperkähler fourfolds in specified conditions.

We prove that the compact Kaehler manifolds with first Chern class nonnegative that admit holomorphic parabolic geometries are the flat bundles of rational homogeneous varieties over complex tori. We also prove that the compact Kaehler manifolds with negative first Chern class that admit holomorphic cominiscule geometr…

Compact pseudo-Hermitian spaces have rigid holomorphic isometries.

Let $M$ be a hyperkähler manifold with $b_2(M)\geq 5$. We improve our earlier results on the Morrison-Kawamata cone conjecture by showing that the Beauville-Bogomolov square of the primitive MBM classes (i.e. the classes whose orthogonal hyperplanes bound the Kähler cone in the positive cone, or, in other words, the cl…

$Γ$-structures are weak forms of multiplications on closed oriented manifolds. As shown by Hopf the rational cohomology algebras of manifolds admitting $Γ$-structures are free over odd degree generators. We prove that this condition is also sufficient for the existence of $Γ$-structures on manifolds which are nilpotent…

The article proves estimates on homotopy and cohomology dimensions in fibrations.

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively cur…

Let $E\to M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$. We prove that if $E$ admits a $ω$-balanced metric (in X. Wang's terminology) then it is unique. This result together with a result of L. Biliotti and A. Ghigi implies the existence and uniqueness of $ω$-balanced metrics of certain dir…

This paper completes the classification of certain surface singularities with rational homology disk smoothings.

We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.

The paper proves conditions for projectivity and rational connectedness of Kähler manifolds.

Classifies degenerations of complex projective plane with rational singularities.

Let $(X,ω)$ be an $n$-dimensional compact Kähler manifold. We study degenerate complex Hessian equations of the form $(ω+dd^c\varphi)^m\wedge ω^{n-m}=F(x,\varphi)ω^n.$ Under some natural conditions on $F$, this equation has a unique continuous solution. When $(X,ω)$ is rational homogeneous we further show that the solu…

Several large classes of homogeneous spaces are known to be formal---in the sense of Rational Homotopy Theory. However, it seems that far fewer examples of non-formal homogeneous spaces are known. In this article we provide several construction principles and characterisations for non-formal homogeneous spaces, which w…

The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculated together with its Pontrjagin structure. Second, it is shown that the integral homology of the…

Generalizes Lefschetz fibrations with rational homology disk smoothings.

Let $E\rightarrow M$ be a holomorphic vector bundle over a compact Kaehler manifold $(M, ω)$ and let $E=E_1\oplus... \oplus E_m\rightarrow M$ be its decomposition into irreducible factors. Suppose that each $E_j$ admits a $ω$-balanced metric in Donaldson-Wang terminology. In this paper we prove that $E$ admits a unique…

The Yamabe invariant is linked to static potentials and eigenvalues.

New findings on minimal isometric immersions of flat n-tori into spheres.

We classify compact 2-connected homogeneous spaces with the same rational cohomology as a product of spheres. This classification relies on spectral sequences, homotopy theory, and representation theory. We then apply this classification to two geometric problems. The first problem is the classification of all isoparam…

New Lie group approach for envelope surface computation.

New dHYM connections found on complex vector bundles.

Study shows infinite order in mapping class groups for certain 3D shapes.