arXiv research

In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.

We show that an $n-$dimensional Moishezon manifold is uniruled if and only if it supports a balanced metric $ω^{n-1}$ of positive total scalar Chern curvature. A similar statement also holds true for class $\mathcal C$ manifolds of dimension three.

Study 1-flat G-structures on uniruled projective manifolds.

Survey on minimal rational curves and their geometric structures.

We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This…

In this note we extend to non trivial Hamiltonian fibrations over symplectically uniruled manifolds a result of Lu's, \cite{Lu}, stating that any trivial symplectic product of two closed symplectic manifolds with one of them being symplectically uniruled verifies the Weinstein Conjecture for closed separating hypersurf…

A symplectic manifold $(M,ω)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symple…

It is a basic tenet in complex geometry that {\it negative} curvature corresponds, in a suitable sense, to the absence of rational curves on, say, a complex projective manifold, while {\it positive} curvature corresponds to the abundance of rational curves. In this spirit, we prove in this note that a projective manifo…

The study proves a partial converse to the Andreotti-Grauert theorem for complex manifolds.

We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate it in terms of Gromov-Witten invariants. The (pseudo) symplectic capacitie…

This paper explores the topology of monotone Lagrangian submanifolds $L$ inside a symplectic manifold $M$ by exploiting the relationships between the quantum homology of $M$ and various quantum structures associated to the Lagrangian $L$.

We classify holomorphic Cartan geometries on every compact complex curve, and on every compact complex surface which contains a rational curve.

This is an expanded and updated version of a lecture series I gave at Seoul National University in September 1997. It is in some sense an update of the 1979 Griffiths and Harris paper with a similar title. I discuss: Homogeneous varieties, Topology and consequences Projective differential invariants, Varieties with deg…

The paper bounds Chern numbers of threefolds, generalizing previous work.

In this paper we aim at the description of foliations having tangent sheaf $T\mathcal F$ with $c_1(T\mathcal F)=c_2(T\mathcal F)=0$ on non-uniruled projective manifolds. We prove that the universal covering of the ambient manifold splits as a product, and that the Zariski closure of a general leaf of $\mathcal F$ is an…

Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.

The article provides obstructions for exact submanifolds in symplectic applications.

Let $φ:\Bbb C^n\to X$ a holomorphic map to an $n$-dimensional connected compact complex manifold $X$. We establish links between the positivity properties of the canonical bundle of $X$ and the rate of growth of $φ$ which extend results of Kodaira and Kobayashi-Ochiai. For example: if the average degree of $φ$ on balls…

In a previous paper, we proved that a projective Kähler manifold of positive total scalar curvature is uniruled. At the other end of the spectrum, it is a well-known theorem of Campana and Kollár-Miyaoka-Mori that a projective Kähler manifold of positive Ricci curvature is rationally connected. In the present work, we …

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 …

The study explores geometric properties of hyperbolic cohomology classes on Kähler manifolds.

Researchers characterize a specific type of projective variety based on its tangents.

Let $(\mathcal{E}, φ)$ be a rank two co-Higgs vector bundles on a Kähler compact surface $X$ with $φ\in H^0(X,End(\mathcal{E})\otimes T_X)$ nilpotent. If $(\mathcal{E}, φ)$ is semi-stable, then one of the following holds up to finite \' etale cover: $i)$ $X$ is uniruled. $ii)$ $X$ is a torus and $(\mathcal{E}, φ)$ is s…

Two types of nonvanishing results are presented for compact Kähler varieties.

Study of rational curves in complex manifolds with specific normal bundles.

Characterizes symplectic and odd-symplectic Grassmannians using VMRT.

The paper explores cone structures and their connections to parabolic geometries in complex manifolds.

New class of metric f-manifolds introduced.

Characterizes and describes selfsimilar Hessian manifolds with homothetic vector fields.

Extends Tian theorem to Vaisman manifolds for approximations.

The paper defines trans-para-Sasakian manifolds and explores their geometric properties.

In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…

Study on a new type of manifolds that generalize almost C-manifolds.

New class of complex manifolds defined, properties studied.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.

Determines the type of 3-manifolds from their covers.

The study identifies criteria for 3-manifolds to be boundaries of exotic 4-manifolds.

The study provides a structure theorem for a new class of noncompact 3-manifolds.

Condition for intersection of real flag manifolds in complex flag manifold.

The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.

Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.

Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.

Study on 3D manifolds with specific tensor structures and their properties.

New manifold type PNDP-manifold defined with Einstein warped product structure.

Constructs higher-dimensional twistor spaces for complex manifolds.

Optimal inequalities for systole, inradius, and volume in hyperbolic 3-manifolds

The paper explores F-manifolds and metrics, constructing canonical structures.