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SUMMARY:Embeddings of smooth affine varieties into linear algebraic groups
DTSTART;VALUE=DATE-TIME:20211014T104500
DTEND;VALUE=DATE-TIME:20211014T104500
UID:65d735ac5e72c8d7a1bd3cfbc9a881d0d6a7cedfe9deac0a9eb9be62
CATEGORIES:Conferences - Seminars
DESCRIPTION:Immanuel van Santen (Unibasel)\nThis is joint work with Peter
Feller. There are the following fundamental questions concerning embedding
s from an object Z into another object X:\n\n 1. (Existence) D
oes there exists an embedding of Z into X?\n 2. (Uniqueness) H
aving two embeddings f\, g of Z into X\, does there exists an automorphism
ψ of X such that g = ψ ◦ f?\n\nIn this talk\, we will mainly focus
on the first question in the category of affine varieties\, where Z is smo
oth and X is a\nlinear algebraic group. Amongst other things\, we will dis
cuss the following result.\n\nTheorem. For every simple linear algebraic g
roup G and every smooth affine variety Z with dim G > 2 dim Z + 1\, there
exists an embedding of Z into G.\n\nThe proof is based upon parametric tr
ansversality results for flexible affine varieties due to Kaliman. We will
also discuss the following result.\n\nProposition. For every non-finite a
lgebraic group G and every d ≥ dim G / 2\, there exists an irreducible s
mooth affine variety of dimension d that does not admit an embedding into
G.\n\nThis proposition implies the optimality of the above existence theor
em up to a possible improvement of the dimension bound by one. It’s proo
f is an adaptation of a Chow-group-based argument due to Bloch\, Murthy\,
and Szpiro for the affine space.
LOCATION:MA B1 11 https://plan.epfl.ch/?room==MA%20B1%2011
STATUS:CONFIRMED
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