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UID:news699@dmi.unibas.ch
DTSTAMP;TZID=Europe/Zurich:20190202T205342
DTSTART;TZID=Europe/Zurich:20150410T103000
SUMMARY:Seminar Algebra and Geometry: Hanspeter Kraft (Universität Basel)
DESCRIPTION:We show that the automorphism group of affinen-space An determi
 nes An up to isomorphism:  If X is a connected affine variety such that Au
 t(X) \\simeq Aut(An) as ind-groups\, then X \\simeq An as varieties.We  al
 so  show  that  every  torus  appears  as  Aut(X)  for  a  suitable  affin
 e  variety X\,  but that  Aut(X)  cannot  be  isomorphic  to  a  semisimpl
 e  group.   In  fact\,  if  Aut(X)  is  finitedimensional and if X \\not\\
 simeq A1\, then the connected component Aut(X)◦ is a torus.Concerning th
 e structure of Aut(An) we prove that any homomorphism Aut(An)→ G of ind-
 groups either factors through jac : Aut(An)→C∗ where jac is the Jacobi
 an determinant\, or it is a closed immersion.  For SAut(An) := ker(jac)⊂
 Aut(An) we show that everynontrivial homomorphism SAut(An)→G is a closed
  immersion.Finally\,  we prove that every non-trivial homomorphism φ: SAu
 t(An)→SAut(An) is anautomorphism\, and that φ is given by conjugation w
 ith an element from Aut(An).
X-ALT-DESC: We show that the automorphism group of affinen-space <b>A</b><s
 up>n</sup> determines <b>A</b><sup>n</sup> up to isomorphism:  If X is a c
 onnected affine variety such that Aut(X) \\simeq Aut(<b>A</b><sup>n</sup>)
  as ind-groups\, then X \\simeq <b>A</b><sup>n</sup> as varieties.<br />We
   also  show  that  every  torus  appears  as  Aut(X)  for  a  suitable  a
 ffine  variety X\,  but that  Aut(X)  cannot  be  isomorphic  to  a  semis
 imple  group.   In  fact\,  if  Aut(X)  is  finitedimensional and if X \\n
 ot\\simeq <b>A</b><sup>1</sup>\, then the connected component Aut(X)<sup>
 ◦ </sup>is a torus.<br />Concerning the structure of Aut(<b>A</b><sup>n<
 /sup>) we prove that any homomorphism Aut(An)→ G of ind-groups either fa
 ctors through jac : Aut(<b>A</b><sup>n</sup>)→C<sup>∗</sup> where jac 
 is the Jacobian determinant\, or it is a closed immersion.  For SAut(<b>A<
 /b><sup>n</sup>) := ker(jac)⊂Aut(<b>A</b><sup>n</sup>) we show that ever
 ynontrivial homomorphism SAut(An)→G is a closed immersion.<br />Finally\
 ,  we prove that every non-trivial homomorphism φ: SAut(<b>A</b><sup>n</s
 up>)→SAut(<b>A</b><sup>n</sup>) is anautomorphism\, and that φ is given
  by conjugation with an element from Aut(<b>A</b><sup>n</sup>).
DTEND;TZID=Europe/Zurich:20150410T120000
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