Algebraic Geometry Seminar

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Isomorphisms between complex algebraic and analytic varieties

Speaker: Herwig Hauser, University of Vienna

Location: TBA

Date: Tuesday, April 15, 2025, 5 p.m.

Synopsis:

Zariski asked whether every isomorphism between polynomial rings A[t]
and B[t] over finitely generated C-algebras comes from an isomorphism
between A and B. Geometrically, does X x A^1 isomorphic to Y x A^1 imply
X isomorphic Y. This is one version of a cancellation theorem, and it is
wide open if one replaces A^1 by higher dimensional varieties.

We will address the same problem in the local context of germs of
complex analytic varieties and biholomorphic maps between them. Does X x
Z \isom Y x Z imply X \isom Y?

The answer is, surprisingly, "Yes", but the proof is hard (though
elementary). More generally, one can prove a factorization theorem X =
X_1 x ... x X_n with unique factors X_i which are not further
decomposable. This requires Popescu's famous nested Artin approximation
theorem.

Interesting is also Aut(X), the (infinite dimensional) group of
holomorphic automorphism of a germ X. It can be seen as the group of
symmetries of X.

Theorem: If Aut(X) \isom Aut(Y) as topological groups, then X \isom Y.

And, finally, we will return to the Zariski topology and algebraic
varieties, namely those which admit an open covering by patches
isomorphic to open subsets of A^n ("plain" or "uniformly rational"
varieties). These are still quite mysterious. However, one can at least
show that they are stable under blowup.