21 research outputs found
Proper twin-triangular Ga-actions on A^4 are translations
An additive group action on an affine 3 -space over a complex Dedekind domain
A is said to be twin-triangular if it is generated by a locally nilpotent
derivation of A[y,z,t] of the form rd/dy+p(y)d/dz + q(y)d/dt, where r belongs
to A and p,q belong to A[y] . We show that these actions are translations if
and only if they are proper. Our approach avoids the computation of rings of
invariants and focuses more on the nature of geometric quotients for such
actions
On exotic affine 3-spheres
Every bundle over the complex affine plane punctured at the
origin, is trivial in the differentiable category but there are infinitely many
distinct isomorphy classes of algebraic bundles. Isomorphy types of total
spaces of such algebraic bundles are considered; in particular, the complex
affine 3-sphere admitts such a structure with an additional homogeneity
property. Total spaces of nontrivial homogeneous -bundles over
the punctured plane are classified up to -equivariant algebraic
isomorphism and a criterion for nonisomorphy is given. In fact the affine
3-sphere is not isomorphic as an abstract variety to the total space of any
-bundle over the punctured plane of different homogeneous
degree, which gives rise to the existence of exotic spheres, a phenomenon that
first arises in dimension three. As a by product, an example is given of two
biholomorphic but not algebraically isomorphic threefolds, both with a trivial
Makar-Limanov invariant, and with isomorphic cylinders
Factorial threefolds with Ga-actions
The affine cancellation problem, which asks whether complex affine varieties
with isomorphic cylinders are themselves isomorphic, has a positive solution
for two dimensional varieties whose coordinate rings are unique factorization
domains, in particular for the affine plane, but counterexamples are found
within normal surfaces Danielewski surfaces and factorial threefolds of
logarithmic Kodaira dimension equal to 1. The latter are therefore remote from
the affine three-space, the first unknown case where the base of one cylinder
is an affine space. Locally trivial Ga-actions play a significant role in these
examples. Threefolds admitting free Ga-actions are discussed, especially a
class of varieties with negative logarithmic Kodaira dimension which are total
spaces of nonisomorphic Ga-bundles. Some members of the class are shown to be
isomorphic as abstract varieties, but it is unknown whether any members of the
class constitute counterexamples to cancellation
Local Triviality of Proper Ga Actions
AbstractRegular actions of the additive group of complex numbers on complex surfaces and on complex affine space are considered. A proper action on an affine surface admits a geometric quotient which is an affine curve. A proper action on a normal quasiaffine surface is equivariantly trivial. New criteria for local and âglobalâ triviality of proper actions on a complex affine space of arbitrary dimension are presented