arXiv research

The paper proves a section for Anosov vector fields on compact manifolds.

The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.

The paper extends a theorem about Kähler manifolds with quasi-negative curvature to almost quasi-negative curvature.

We show that if a compact complex manifold admits a Kähler metric whose holomorphic sectional curvature is everywhere non positive and strictly negative in at least one point, then its canonical bundle is positive.

Verify conjecture for special Hermitian manifolds.

We prove that a smooth complex projective threefold with a Kähler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, …

We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the pro…

Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.

We prove that any $4$-dimensional almost-Kähler Lie algebra of constant Hermitian holomorphic sectional curvature with respect to the canonical Hermitian connection is Kähler.

In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

The paper shows how different geodesic flows on surfaces can be mapped to each other.

We prove the existence of a (unique) S^1-invariant Ricci-flat Kaehler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kaehler manifold X, extending the metric on X.

Two remarks on curvature properties of Kähler manifolds.

We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors, and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to geometrically realize these results to produce a hypersurface with prescribed sectional curv…

Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact Kähler manifolds. In this short note, we give anothe proof of their theorems by using the Kähler-Ricci flow.

We prove that a simpy connected Hermitian Einstein 4-manifold with non-negative sectional curvature is isometric to complex projective space $\mathbb{C}\mathbb{P}^{2}$ with the Fubini-Study metric or isometric to the product $\mathbb{S}^{2}\times \mathbb{S}^{2}$ with the canonical metric.

We consider natural differential operations acting on sections of tensor vector bundles. Arrising problems can be reformulated as invariant theoretical problems (the IT-reduction). We give examples of usage of the IT-reduction. In particular, on a manifold with a connection and a Poisson structure we construct the cano…

Proves a conjecture for Calabi-Yau manifolds.

In this paper, we give some estimates of the sum of the square norm of the sections of the pluricanonical bundles over a Riemann surface with genus greater than 2 and Gauss curvature (-1). Using these estimate, we give a uniform estimate of the corona problem on Riemann surfaces.

Geometrically proves Zabrodin-Wiegmann conjecture for integer QH states.

Given any compact Riemann surface $C$, there is a canonical meromorphic 2--form $\widehatη$ on $C\times C$, with pole of order two on the diagonal $Δ\, \subset\, C\times C$, constructed in \cite{cfg}. This meromorphic 2--form $\widehatη$ produces a canonical projective structure on $C$. On the other hand the uniformiza…

Harmonic unit normal sections studied for Grassmannians induced by cross products.

This paper studies the Fisher-Rao geometry on the parameter space of beta distributions. We derive the geodesic equations and the sectional curvature, and prove that it is negative. This leads to uniqueness for the Riemannian centroid in that space. We use this Riemannian structure to study canonical moments, an intrin…

On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional…

Gradient and eigenvalue estimates for Kähler manifolds' canonical bundle.

Symmetric spaces' connections form Lie admissible triple algebras.

The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold $M$. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …

When a gauge-natural invariant variational principle is assigned, to determine {\em canonical} covariant conservation laws, the vertical part of gauge-natural lifts of infinitesimal principal automorphisms -- defining infinitesimal variations of sections of gauge-natural bundles -- must satisfy generalized Jacobi equat…

A new method for clearing liability networks using sheaves on directed hypergraphs.

The paper classifies translation surfaces with constant curvature in a specific connection.

Characterizes complex Finsler metrics and their properties.

Discrete analogues of ellipsoids with preserved circular cross sections.

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a g…

Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.

A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canon…

Study proves properties of compact Hermitian surfaces with specific curvature conditions.

The study defines a canonical nilpotent structure for certain collapsed manifolds.

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

Theory of T-duality for transitive Courant algebroids developed.

Let $(M,ω)$ be a compact Kähler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is necessarily projective and has ample canonical bundle. In this paper, we show that any irreducible subvariety of $M$ is of general type. Moreover, we can extend the theorem to the…

Upper bound found for dimensions of subspaces where holomorphic sectional curvature vanishes.

Study shows Calabi-Yau metrics converge to a specific form under certain conditions.

In recent papers Wu-Yau, Tosatti-Yang and Diverio-Trapani, used some natural differential inequalities for compact Kähler manifolds with quasi negative holomorphic sectional curvature to derive positivity of the canonical bundle. In this note we study the equality case of these inequalities.

In the first part of this paper we consider compact algebraic manifolds M^2n with an algebraic (n-1)-Torus action. We show that there is a T-invariant meromorphic section $σ$ of the canonical bundle of M. Any such $σ$ defines a divisor D. On the complement M'=M-D we have a trivialization of the canonical bundle and a T…

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then t…

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature $k\geq 0$ with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observa…

Study on surfaces with constant curvature under a specific connection.

We show that the canonical central extension of the group of sections of a Lie group bundle over a compact manifold, constructed in [NW09], is universal. In doing so, we prove universality of the corresponding central extension of Lie algebras in a slightly more general setting.