Flexible varieties and automorphism groups

By Ivan Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, Mikhail Zaidenberg. 2010

185 citations recorded by the source corpus.

Given an irreducible affine algebraic variety X of dimension n≥2 , we let SAut(X) denote the special automorphism group of X , that is, the subgroup of the full automorphism group Aut(X) generated by all one-parameter unipotent subgroups. We show that if SAut(X) is transitive on the smooth locus X reg , then it is infinitely transitive on X reg . In turn, the transitivity is equivalent to the flexibility of X . The latter means that for every smooth point x∈X reg the tangent space T x X is spanned by the velocity vectors at x of one-parameter unipotent subgroups of Aut(X) . We also provide various modifications and applications.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6efaf7a95077faaf5d23e591bd545d352a30de25

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In Geometry and Topology

Citations in the corpus

Holomorphic Approximation: The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan (2018, background): “s manifolds, but also many nonhomogeneousones. For example, if the tangent bundleTY of a complexmanifoldY is pointwisegeneratedbyC-completeholomorphicvectorﬁelds on Y (such manifold is called ﬂexible [9]), thenY is an Oka manifold [66, Proposition 5.6.22].For examples and properties of Oka manifolds”
Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5 (2011, unknown): “Then the following conditions are equivalent : (1) the variety X is flexible; (2) the group SAutX acts transitively on the smooth locus Xreg of X ; (3) the group SAutX acts infinitely transitively on Xreg .”