arXiv subjects
Publications and source records attributed to Rafael B. Andrist.
We prove that the bracket width of the Lie algebra of vector fields on any smooth affine algebraic variety of dimension $n$ is at most $(n+1)^2$. We give improved bounds for some families of $\mathbb{C}^*$-varieties, in particular for $\mathrm{SL}_n(\mathbb{C})$ and for the Koras--Russell cubic threefold.
Let $B$ be an affine $k$-domain which admits a nontrivial fundamental pair $(D,U)$ of locally nilpotent derivations, i.e., if $E=[D,U]$ then $(D,U,E)$ is an $\mathfrak{sl}_2$-triple. We prove an algebraic criterion, characterizing under which conditions the fundamental pair $(D,U)$ resp. the triple $(D,U,E)$ is compatible in a technical sense that allows us to construct many vector fields on the spectrum of $B$ from the complete ones. This criterion enables us to prove the algebraic density property for the following widely studied classes of $\mathrm{SL}_2$-varieties arising in physics: Classical Calogero--Moser spaces, Calogero--Moser spaces with "inner degrees of freedom'' and a smooth cyclic quiver variety.
We prove that for many pairs $H_1, H_2$ of root subgroups of the automorphism group $\text{Aut}(\mathbb{C}^2)$ the diagonal action of the group generated by $H_1, H_2$ on $(\mathbb{C}^2)^m$ has an open orbit for any positive integer $m$. The result is based on the study of the Lie algebra of polynomials in two variables with the standard Poisson bracket.
We show that pseudoconvex Reinhardt domains in dimension two with isomorphic semigroups of holomorphic endomorphisms are biholomorphically or anti-biholomorphically equivalent. Moreover, we show that every Stein manifold that retracts to a properly embedded copy of the punctured complex line, is determined (up to biholomorphic or anti-biholomorphic equivalence) by its semigroup of holomorphic endomorphisms.
We study the Lie algebra of polynomial vector fields on a smooth Danielewski surface of the form $x y = p(z)$ with $x,y,z \in \mathbb{C}$. We provide explicitly given generators to show that: 1. The Lie algebra of polynomial vector fields is generated by $6$ complete vector fields. 2. The Lie algebra of volume-preserving polynomial vector fields is generated by finitely many vector fields, whose number depends on the degree of the defining polynomial. 3. There exists a Lie sub-algebra generated by $4$ LNDs whose flows generate a group that acts infinitely transitively on the Danielewski surface. The latter result is also generalized to higher dimensions where $z \in \mathbb{C}^N$.
We show that the direct product of two Stein manifolds with the Hamiltonian density property enjoys the Hamiltonian density property as well. We investigate the relation between the Hamiltonian density property and the symplectic density property. We then establish the Hamiltonian and the symplectic density property for $(\mathbb{C}^\ast)^{2n}$ and for the so-called traceless Calogero--Moser spaces. As an application we obtain a Carleman-type approximation for Hamiltonian diffeomorphisms of a real form of the traceless Calogero--Moser space.
Every complex surface of Markov type, i.e.\ the variety given by $x^2 + y^2 + z^2 + Exyz - Ax - By - Cz - D = 0$, has the symplectic density property and the Hamiltonian density property. We prove a singular symplectic version of the Anders{é}n--Lempert theorem for normal reduced affine complex varieties and apply it to describe the holomorphic symplectic automorphisms of a complex surface of Markov type. To this end, we also investigate the germs of vector fields in isolated singularities of type $A_k$ and $D_k$. Moreover, we show that any injective self-map of the set of ordered Markov triples can be realized by a holomorphic symplectic automorphism.
We prove the density property for generalized Calogero--Moser spaces with inner degrees of freedom. This allows us to describe the holomorphic automorphism group of these complex affine manifolds. These generalized Calogero--Moser spaces can also be understood as quiver varieties corresponding to moduli spaces of $\mathrm{SU}(2)$ instantons on a non-commutative $\mathbb{R}^4$.
We prove a parametric jet interpolation theorem for symplectic holomorphic automorphisms of $\mathbb{C}^{2n}$ with parameters in a Stein space. Moreover, we provide an example of an unavoidable set for symplectic holomorphic maps.
The boundary of every relatively compact Stein domain in a complex manifold of dimension at least two is connected. No assumptions on the boundary regularity are necessary. The same proofs hold also for $q$-complete domains, and in the context of almost complex manifolds as well.
We introduce the notion of the algebraic overshear density property which implies both the algebraic notion of flexibility and the holomorphic notion of the density property. We investigate basic consequences of this stronger property, and propose further research directions in this borderland between affine algebraic geometry and elliptic holomorphic geometry. As an application, we show that any smoothly bordered Riemann surface with finitely many boundary components that is embedded in a complex affine surface with the algebraic overshear density property admits a proper holomorphic embedding.
We generalize a criterion for the density property of Stein manifolds. As an application, we give a new, simple proof of the fact that the Danielewski surfaces have the algebraic density property. Furthermore, we have found new examples of Stein manifolds with the density property.
We introduce the symplectic holomorphic density property and the Hamiltonian holomorphic density property together with the corresponding version of Andersén-Lempert theory. We establish these properties for the Calogero-Moser space $\mathcal{C}_n$ of $n$ particles and describe its group of holomorphic symplectic automorphisms.
For the special linear group $\mathrm{SL}_2(\mathbb{C})$ and for the singular quadratic Danielewski surface $x y = z^2$ we give explicitly a finite number of complete polynomial vector fields that generate the Lie algebra of all polynomial vector fields on them. Moreover, we give three unipotent one-parameter subgroups that generate a subgroup of algebraic automorphisms acting infinitely transitively on $x y = z^2$.
We prove the existence of strongly tame sets in affine algebraic homogenenous spaces of linear algebraic Lie groups. We also show that $(\mathbb{C}^n,A)$ for a discrete tame set enjoy the relative density property, and we provide examples of Stein manifolds admitting non-equivalent tame sets.
We prove the algebraic density property for the Calogero--Moser spaces $\mathcal{C}_{n}$, and give a description of the identity component of the group of holomorphic automorphisms of $\mathcal{C}_{n}$.
There exist three vector fields with complete polynomial flows on $\mathbb{C}^n$, $n \geq 2$, which generate the Lie algebra generated by all algebraic vector fields on $\mathbb{C}^n$ with complete polynomial flows. In particular, the flows of these vector fields generate a group that acts infinitely transitive. The analogous result holds in the holomorphic setting.
We generalize the notion of tame discrete sets introduced by Rosay and Rudin from complex-Euclidean space to arbitrary complex manifolds and establish their basic properties. We show that complex-linear algebraic groups different from the complex line or the punctured complex line contain tame discrete sets.