Consider a Cremona group endowed with the Euclidean topology introduced by Blanc and Furter. It makes it a Hausdorff topological group that is not locally compact nor metrisable. We show that any sequence of elements of the Cremona group of bounded order that converges to the identity is constant. We use this result to show that the Cremona groups do not contain any non-trivial sequence of subgroups converging to the identity. We also show that, in general, paths in a Cremona group do not lift and do not satisfy a property similar to the definition of morphisms to a Cremona group.
Abstract. Let M be a compact complex supermanifold. We prove that the set Aut0(M) of automorphisms of M can be endowed with the structure of a complex Lie group acting holomorphically on M, so that its Lie algebra is isomorphic to the Lie algebra of even holomorphic super vector fields on M. Moreover, we prove the existence of a complex Lie supergroup Aut(M) acting holomorphically on M and satisfying a universal property. Its underlying Lie group is Aut0(M) and its Lie superalgebra is the Lie superalgebra of holomorphic super vector fields on M. This generalizes the classical theorem by Bochner and Montgomery that the automorphism group of a compact complex manifold is a complex Lie group. Some examples of automorphism groups of complex supermanifolds over P1(C) are provided.
We prove the Lipman-Zariski conjecture for complex surface singularities of genus at most two, and also for those of genus three whose link is not a rational homology sphere. This improves on a previous result of the second author. As an application, we show that a compact complex surface with locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.
We consider a version of the Lipman-Zariski conjecture for logarithmic vector fields and logarithmic 1-forms on pairs. Let (X, D) be a pair consisting of a normal complex variety X and an effective Weil divisor D such that the sheaf of logarithmic vector fields (or dually the sheaf of reflexive logarithmic 1-forms) is locally free. We prove that in this case the following holds: If (X, D) is dlt, then X is necessarily smooth and ⌊D⌋ is snc. If (X, D) is lc or the logarithmic 1-forms are locally generated by closed forms, then (X, ⌊D⌋) is toroidal.
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