Related papers: Limit points and additive group actions

We studied an enhanced adjoint action of the general linear group on a product of its Lie algebra and a vector space consisting of several copies of defining representations and its duals. We determined regular semisimple orbits (i.e.,…

Let G be an n-dimensional torus and $\tau$ a Hamiltonian action of G on a compact symplectic manifold, M. If M is pre-quantizable one can associate with $\tau$ a representation of G on a virtual vector space, Q(M), by…

In this note we study the problem of characterizing the complex affine space $\mathbb{A}^n$ via its automorphism group. We prove the following. Let $X$ be an irreducible quasi-projective $n$-dimensional variety such that $\mathrm{Aut}(X)$…

We associate a root system to a finite set in a free abelian group and prove that its irreducible subsystem is of type A, B or D. We apply this general result to a torus manifold, where a torus manifold is a $2n$-dimensional connected…

Let $G$ be a connected algebraic $k$-group acting on a normal $k$-variety, where $k$ is a field. We show that $X$ is covered by open $G$-stable quasi-projective subvarieties; moreover, any such subvariety admits an equivariant embedding…

We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on $X_{K'}$. In…

Let $X$ be a compact smooth manifold, possibly with boundary. Denote by $X_1,\dots,X_r$ the connected components of $X$. Assume that the integral cohomology of $X$ is torsion free and supported in even degrees. We prove that there exists a…

Let $X$ be a Nakajima quiver variety and $X'$ its $3d$-mirror. We consider the action of the Picard torus $\mathsf{K}=\mathrm{Pic}(X)\otimes \mathbb{C}^{\times}$ on $X'$. Assuming that $(X')^{\mathsf{K}}$ is finite, we propose a formula for…

We study the action of a finite group on the Riemann-Roch space of certain divisors on a curve. If $G$ is a finite subgroup of the automorphism group of a projective curve $X$ over an algebraically closed field and $D$ is a divisor on $X$…

This paper constructs derived autoequivalences associated to an algebraic flopping contraction \(X\to X_{\con}, \) where \(X\) is quasi-projective with only mild singularities. These functors are constructed naturally using bimodule cones,…

The main goal of this paper is to obtain a formula for the T-equivariant Riemann-Roch number of certain G-spaces which are the finite dimensional models of certain infinite dimensional spaces with Hamiltonian LG-actions, here T is a maximal…

Let $\mathcal{G}$ be an ind-group and let $\mathcal{U} \subseteq \mathcal{G}$ be a unipotent ind-subgroup. We prove that an abstract group automorphism $\theta \colon \mathcal{G} \to \mathcal{G}$ maps $\mathcal{U}$ isomorphically onto a…

Let G be an affine algebraic group and let X be an affine algebraic variety. An action $G\times X \to X$ is called observable if for any G-invariant, proper, closed subset Y of X there is a nonzero invariant $f\in K[X]^G$ such that f(Y) =0.…

We prove: Let $G$ be a finite abelian group acting faithfully on a complex smooth project variety $X$ of general type with numerically effective canonical divisor, of dimension $n$. Then $$|G| \le C(n)K_X^n ,$$ where $C(n)$ depends only on…

We study complete, connected and simply connected $n$-dim Riemannian manifold $M$ satisfying Ricci curvature lower bound. Further more, suppose that $M$ admits discrete isometric group actions $G$ so that the diameter of the quotient space…

We show that a compact Kaehler manifold X is a complex torus if both the continuous part and discrete part of some automorphism group G of X are infinite groups, unless X is bimeromorphic to a non-trivial G-equivariant fibration. Some…

If $G$ is a compact Lie group endowed with a left invariant metric $g$, then $G$ acts via pullback by isometries on each eigenspace of the associated Laplace operator $\Delta_g$. We establish algebraic criteria for the existence of left…

Let $k$ be a finitely generated field, let $X$ be an algebraic variety and $G$ a linear algebraic group, both defined over $k$. Suppose $G$ acts on $X$ and every element of a Zariski-dense semigroup $\Gamma \subset G(k)$ has a rational…

Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $\pi_X:X\to X/G$. We characterize rational tensor fields $\tau$ on $X/G$ such that the {\it pull back} of $\tau…

Consider a smooth action $\mathbf G\times M \rightarrow M$ of a compact connected Lie group $\mathbf G$ on a connected manifold $M$. Assume the existence of a point of $M$ whose isotropy group has a single element (free point). Then we…