We prove that a symplectic matrix with entries in a ring with Bass stable rank one can be factored as a product of elementary symplectic matrices. This also holds for null-homotopic symplectic ...matrices with entries in a Banach algebra or in the ring of complex valued continuous functions on a finite dimensional normal topological space.
We enlarge the class of open Riemann surfaces known to be holomorphically embeddable into the plane by allowing them to have additional isolated punctures compared to the known embedding results.
Homotopy principles for equivariant isomorphisms Kutzschebauch, Frank; Lárusson, Finnur; Schwarz, Gerald
Transactions of the American Mathematical Society,
10/2017, Volume:
369, Issue:
10
Journal Article
Peer reviewed
Open access
Let G be a reductive complex Lie group acting holomorphically on Stein manifolds X and Y. Let p_X\colon X\to Q_X and p_Y\colon Y\to Q_Y be the quotient mappings. When is there an equivariant ...biholomorphism of X and Y? A necessary condition is that the categorical quotients Q_X and Q_Y are biholomorphic and that the biholomorphism \phi sends the Luna strata of Q_X isomorphically onto the corresponding Luna strata of Q_Y. Fix \phi . We demonstrate two homotopy principles in this situation. The first result says that if there is a G-diffeomorphism \Phi \colon X\to Y, inducing \phi , which is G-biholomorphic on the reduced fibres of the quotient mappings, then \Phi is homotopic, through G-diffeomorphisms satisfying the same conditions, to a G-equivariant biholomorphism from X to Y. The second result roughly says that if we have a G-homeomorphism \Phi \colon X\to Y which induces a continuous family of G-equivariant biholomorphisms of the fibres p_X{^{-1}}(q) and p_Y{^{-1}}(\phi (q)) for q\in Q_X and if X satisfies an auxiliary property (which holds for most X), then \Phi is homotopic, through G-homeomorphisms satisfying the same conditions, to a G-equivariant biholomorphism from X to Y. Our results improve upon those of our earlier paper J. Reine Angew. Math. 706 (2015), 193-214 and use new ideas and techniques.
It is an elementary fact that the action by holomorphic automorphisms on
C
n
is infinitely transitive, i.e.,
m
-transitive for any
m
∈
N
. The same holds on any Stein manifold with the holomorphic ...density property
X
. We study a parametrized case: we consider
m
points on
X
parametrized by a Stein manifold
W
, and seek a family of automorphisms of
X
, parametrized by
W
, putting them into a standard form which does not depend on the parameter. This general transitivity is shown to enjoy an Oka principle, to the effect that the obstruction to a holomorphic solution is of a purely topological nature. In the presence of a volume form and of a corresponding density property, similar results for volume-preserving automorphisms are obtained.
Gizatullin surfaces completed by a zigzag of type 0,0,-r_2,-r_3 can be described by the equations yu=xP(x), xv=uQ(u) and yv=P(x)Q(u) in \mathbb{C}^4_{x,y,u,v} where P and Q are non-constant ...polynomials. We establish the algebraic density property for smooth Gizatullin surfaces of this type. Moreover we also prove the density property for smooth surfaces given by these equations when P and Q are holomorphic functions.
We estimate the number of unipotent elements needed to factor a null-homotopic holomorphic map from a finite dimensional reduced Stein space X SL _2(\mathbb{C})
We prove that any Lie subgroup G (with finitely many connected components) of an infinite-dimensional topological group
which is an amalgamated product of two closed subgroups can be conjugated to ...one factor. We apply this result to classify Lie group actions on Danielewski surfaces by elements of the overshear group (up to conjugation).
We define the notion of shears and overshears on a Danielewski surface. We show that the group generated by shears and overshears is dense (in the compact open topology) in the path-connected ...component of the identity of the holomorphic automorphism group.
Let
X
be a connected affine homogenous space of a linear algebraic group
G
over
C
. (1) If
X
is different from a line or a torus we show that the space of all algebraic vector fields on
X
coincides ...with the Lie algebra generated by complete algebraic vector fields on
X
. (2) Suppose that
X
has a
G
-invariant volume form
ω
. We prove that the space of all divergence-free (with respect to
ω
) algebraic vector fields on
X
coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on
X
(including the cases when
X
is a line or a torus). The proof of these results requires new criteria for algebraic (volume) density property based on so called module generating pairs.
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