arXiv research

Study calculates global sections on complex curves.

The study extends cobordism theory to complex sections, defining and calculating cobordism groups.

New method constructs Birkhoff sections for pseudo-Anosov flows with controlled complexity.

Study describes global sections of chiral de Rham complex on compact Ricci-flat Kähler manifolds.

Study shows normal distribution in divisor counts of random sections on complex manifolds.

The paper explores conditions for sections in Lefschetz fibrations and bundles over 2-complexes.

This paper studies torsion obstructions to complex sections on manifolds.

Classifies surfaces of section for Seifert fibrations.

Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.

In this paper, we get an inequality in terms of holomorphic sectional curvature of complex Finsler metrics. As applications, we prove a Schwarz Lemma from a complete Riemannian manifold to a complex Finsler manifold. We also show that a strongly pseudoconvex complex Finsler manifold with semi-positive but not identical…

We prove that a product complex manifold cannot admit a complete Kähler metric with sectional curvature $K<c<0$ and Ricci curvature $Ric > d$, where $c$ and $d$ are constants. In particular, a product domain in $\C$ cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe …

Classifies Nil 3-manifolds as cross-sections of complex hyperbolic surfaces.

The paper proves a Schwarz lemma for mappings between specific types of manifolds.

We consider the local deformation problem of coisotropic submanifolds inside Poisson manifolds. To this end the groupoid of coisotropic sections (with respect to some tubular neighbourhood) is introduced. Although the geometric content of this groupoid is evident, it is usually a very intricate object. We provide a des…

Study of section conjecture analogues over complex numbers and Kodaira fibrations.

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

We find the entropy's infinite-size behavior in complex manifold sections.

We study the space of holomorphic discs with boundary on a surface in a real 2-dimensional vector bundle over a compact 2-manifold. We prove that, if the ambient 4-manifold admits a fibre-preserving transitive holomorphic action, then a section with a single complex point has $C^{2,α}$-close sections such that any (non…

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

Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.

In this paper, we show that every harmonic map from a compact Kähler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact Kähler manifold with positive holomorphic sectional c…

We examine a condition on a simply connected 2-complex X ensuring that groups acting properly on X are coherent. This extends earlier work on 2-complexes with negative sectional curvature which covers the case that G acts freely. Our extension of these results involves a generalization of the notion of sectional curvat…

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the nor…

Researchers confirm conjecture for complex nilmanifolds in higher dimensions.

The paper proves properties of complex manifolds with nonnegative holomorphic sectional curvature.

Suppose that $S^n$ is given a generic Riemannian metric with sectional curvatures which satisfy a suitable pinching condition formulated in terms of complex sectional curvatures. This pinching condition is satisfied by manifolds whose real sectional curvatures $K_r(σ)$ satisfy $1/2 < K_r(σ) \leq 1.$ Then the number o…

Study of random sections on complex spaces converging to equilibrium metrics.

In this paper we will prove that the only compact 4-manifold M with an Einstein metric of positive sectional curvature which is also hermitian with respect to some complex structure on M, is the complex projective plane CP^2, with its Fubini-Study metric.

The Chern sectional curvature of a Hermitian manifold is derived and related to Kähler metrics.

Ricci flow preserves positive sectional curvature on homogeneous spheres

The space of the global sections of chiral de Rham complex on a compact Ricci-flat Kähler manifold is calculated and it is expressed as an invariant subspace of a $βγ-bc$ system under the action of certain Lie algebra.

Invariant complex structures on the homogeneous manifold $U(n+1)/U(n)\times U(p+1)/U(p)$ are reseached. Extreme values of sectional curvature of Hermitian metrics on this manifold are found.

Formula for sections on complex manifolds with non-isolated components.

The Kaehler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kaehler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizi…

Study almost complex structures on six-manifolds using twistor spaces.

Estimates holomorphic sections on Bohr-Sommerfeld Lagrangian submanifolds.

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, …

The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real …

Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.

The study characterizes symmetries in Kaehler manifolds.

We show that there is a complex structure on the symplectic 4-manifold $W_{4, k}$ obtained from the elliptic surface E(4) by rationally blowing down $k$ sections for $2\le k\le 9$. And we interpret it via ${\mathbb Q}$-Gorenstein smoothing. This answers affirmatively to a question raised by R. Gompf.

It is proved that if an almost Kähler manifold of dimension greater or equal to 8 is of pointwise constant antiholomorphic sectional curvature, then it is a complex space form.

New symplectic forms derived from Lagrangian fibrations on symplectic manifolds.

Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Riemannian manifold and prove curvature identities which imply the harmonicity of their parametrizin…

New category theory for complex projective plane sections.

Let $(M,g)$ be a complete noncompact non-collapsing $n$-dimensional riemannian manifold, whose complex sectional curvature is bounded from below and scalar curvature is bounded from above. Then ricci flow with above as its initial data, has at most one solution in the class of complete riemannian metric with complex se…

The following theorem is proved: If an AH3-manifold M of dimension greather or equal to 6 is of pointwise constant antiholomorphic sectional curvature, then M is a real space form or a complex space form.

We find a new relation among right-handed Dehn twists in the mapping class group of a $k$-holed torus for $4 \leq k \leq 9$. This relation induces an elliptic Lefschetz pencil structure on the four-manifold \cp $#(9-k)$ \cpb $$ with $k$ base points and twelve singular fibers. By blowing up the base points we get an el…