Related papers: Zero sets of Abelian Lie algebras of vector fields

Let X be an analytic vector field on a real or complex 2-manifold, and K a compact set of zeros of X whose fixed point index is not zero. Let A denote the Lie algebra of analytic vector fields Y on M such that at every point of M the values…

Let Y and X denote C^k vector fields on a possibly noncompact surface with empty boundary, k >0. Say that Y tracks X if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of…

On a real ($\mathbb F=\mathbb R$) or complex ($\mathbb F=\mathbb C$) analytic connected 2-manifold $M$ with empty boundary consider two vector fields $X,Y$. We say that $Y$ {\it tracks} $X$ if $[Y,X]=fX$ for some continuous function…

A vector field X on a manifold M with possibly nonempty boundary is inward if it generates a unique local semiflow $\Phi^X$. A compact relatively open set K in the zero set of X is a block. The Poincar\'e-Hopf index is generalized to an…

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an…

For a smooth manifold $X$ equipped with a volume form, let $\dL$ be the Lie algebra of volume preserving smooth vector fields on $X$. A. Lichnerowicz proved that the abelianization of $\dL$ is a finite-dimensional vector space, and that its…

The existence of a nowhere zero real vector field implies a well-known restriction on a compact manifold. But all manifolds admit nowhere zero complex vector fields. The relation between these observations is clarified.

A finite dimensional filiform K-Lie algebra is a nilpotent Lie algebra g whose nil index is maximal, that is equal to dim g -1. We describe necessary and sufficient conditions for a filiform algebra over an algebraically closed field of…

We show that if a countable discrete group acts properly and isometrically on a spin manifold of bounded Riemannian geometry and uniformly positive scalar curvature, then, under a suitable condition on the group action, the maximal higher…

It is proven that a local Lie algebra in the sense of A. A. Kirillov determines the base manifold up to a diffeomorphism provided the anchor map is nowhere-vanishing. In particular, the Lie algebras of nowhere-vanishing Poisson or Jacobi…

Results about the following classes of finite-dimensional Lie algebras over a field of characteristic zero are presented: anisotropic (i.e., Lie algebras for which each adjoint operator is semisimple), regular (i.e., Lie algebras in which…

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free…

Let $X$ be an affine algebraic variety with a transitive action of the algebraic automorphism group. Suppose that $X$ is equipped with several non-degenerate fixed point free $SL_2$-actions satisfying some mild additional assumption. Then…

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…

Let $A$ be an algebra over a field $F$ with {\rm char}$(F)\ne 2$. If $A$ is generated as an algebra by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ is an arbitrary vector space over $F$, the…

The existence of a vector field on a compact Kaehler manifold with nonempty zero locus and the properties of this zero locus strongly influence the geometry of the manifold. For example, J. Wahl proved that the existence of a vector field…

A non-associative algebra over a field $\mathbb{K}$ is a $\mathbb{K}$-vector space $A$ equipped with a bilinear operation \[ {A\times A\to A\colon\; (x,y)\mapsto x\cdot y=xy}. \] The collection of all non-associative algebras over…

Let $A$ be a unital associative algebra over a field $F$ and $V$ be a unital left $A$-module. The module $V$ is called zero action determined if every bilinear map $f: A\times V\rightarrow F$ with the property that $f(a,m)=0$ whenever…

A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maximal…

We prove that if $B$ is a $k$-positive holomorphic line bundle on a compact hyperk\"ahler manifold $M,$ then $H^p (M,\Omega^q\otimes B)=0$ for $p>n+[\frac{k}{2}]$ and any nonnegative integer $q.$ In a special case $k=0$ and $q=0$ we recover…