arXiv research

In this paper an analytic proof of a generalization of a theorem of Bismut ([Bis1, Theorem 5.1]) is given, which says that, when $v$ is a transversal holomorphic vector field on a compact complex manifold $X$ with a zero point set $Y$, the embedding $j:Y\to X$ induces a natural isomorphism between the holomorphic equiv…

Study on stability of Sasaki structures under deformations.

Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field

Study of coupled Sasaki-Einstein and solitons metrics.

Let F be a codimension one singular holomorphic foliation on a compact complex manifold M. Assume that there exists a meromorphic vector field X on M generically transversal to F. Then, we prove that F is the meromorphic pull-back of an algebraic foliation on an algebraic manifold N, or F is transversely projective out…

The paper explores how vector fields relate to volume in geometric contexts.

Study curvature lines of a vector field on surfaces.

Study describes lcK structures on Vaisman-type manifolds with holomorphic Lee vector field.

Paper describes holomorphic polyvector fields on toric varieties.

The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.

Constructs the moduli space of super J-holomorphic curves.

Conformal vector fields on lcK manifolds are shown to be Killing or holomorphic.

Classifies vector fields in the kernel of a 1-form, up to equivalence.

The study proves rationality of complex projective varieties with holomorphic vector fields.

We consider vector fields on knot/link complements in $S^3$ which are transverse to the fibres of a fibration of the complement over a circle. We prove that a large class of fibred knots/links, including all non-torus fibred 2-bridge knots, has the following property: any vector field transverse to the fibres of the fi…

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can b…

Holomorphic vector fields and anti-canonical divisors on complex manifolds are studied.

Classifies singularities of smooth vector fields on the line.

The paper studies variations of the Godbillon--Vey invariant for certain foliations.

Study finds conditions for minimum of Donaldson's functional on Higgs bundles with non-holomorphic Higgs fields.

We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.

Defines quaternionic k-vector fields on quaternionic Kähler manifolds.

Study Kähler-Ricci flow on manifolds with singularities.

Explains complex analytic invariants of vector fields and foliations.

The goal of this paper is to study the theory of last multipliers in the framework of complex manifolds with a fixed holomorphic volume form. The motivation of our study is based on the equivalence between a holomorphic ODE system and an associated real ODE system and we are interested how we can relate holomorphic las…

The main theorem of this paper is a result of estimated transversality with respect to stratifications of jet spaces in the approximately holomorphic category over an almost-complex manifold. The notion of asymptotic ampleness of complex vector bundles over an almost-complex manifold is also discussed, as well as appli…

It is well-known that if $ξ$ is a smooth vector field on a given Riemannian manifold $M^n$ then $ξ$ naturally defines a submanifold $ξ(M^n)$ transverse to the fibers of the tangent bundle $TM^n$ with Sasaki metric. In this paper, we are interested in transverse totally geodesic submanifolds of the tangent bundle. We sh…

We classify simply connected compact Sasaki manifolds of dimension $2n+1$ with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture a…

Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.

Study on complex tori foliations and flat geometries.

Holomorphic residue formula for complex supermanifolds.

Compute local cohomology of vector fields on manifolds.

Study vector fields with complex singularities, proving bounds and formulas.

The main result of this paper is the computation of the Lie superalgebras of holomorphic vector fields on complex flag supermanifolds, introduced by Yu.I.Manin. We prove that with several exceptions any holomorphic vector field is fundamental with respect to the natural action of the Lie superalgebra $\mathfrak {gl}_{m…

Flat holomorphic connections on stable bundles over LVMB manifolds are always flat.

Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincare co…

Study BV operators on holomorphic polyvector fields on toric varieties.

Study finds critical points of volume functionals on Sasaki manifolds.

Study of instantons on Sasakian 7-manifolds using gauge fields.

We prove a theorem which asserts that the Lie algebra of all holomorphic vector fields on a compact Kähler manifold with a perturbed extremal metric has the structure similar to the case of an unperturbed extremal Kähler metric proved by Calabi.

We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which…

The Wu-Yau theorem is proven for Sasakian manifolds with specific curvature conditions.

A vector field on a Riemannian manifold is called geodesic if its integral curves are reparametrized geodesics. We classify compact Kähler manifolds admitting nontrivial real-holomorphic geodesic gradient vector fields that satisfy an additional integrability condition. They are all biholomorphic to bundles of complex …

Study of flows on complex manifolds with holomorphic properties.

Establishes jet transversality for regular maps from flexible manifolds.

Study transverse $J$-holomorphic curves linking nearly Kähler $\mathbb{CP}^3$ to minimal surfaces.

We will prove the equivariant version of Smale's transversality theorem: suppose that the compact Lie-group G acts on the compact differentiable manifold M on which an invariant Morse-function f and an invariant vector field X are given so that X is gradient-like with respect to f (i.e. X(f)<0 away from critical orbits…

Affine vector fields on pseudo-Kähler manifolds are symplectic.