Anabelian geometry with étale homotopy types

Abstract

Anabelian geometry with étale homotopy types generalizes in a natural way classical anabelian geometry with étale fundamental groups. We show that, both in the classical and the generalized sense, any point of a smooth variety over a field $k$ that is finitely generated over $\mathbb{Q}$ has a fundamental system of (affine) anabelian Zariski-neighborhoods. This was predicted by Grothendieck in his letter to Faltings.

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      title = {The {G}rothendieck conjecture for affine curves},
      year = {1997},
      pages = {135--194},
      }
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    @BOOK{Whitehead, mrkey = {0516508},
      number = {61},
      author = {Whitehead, George W.},
      mrclass = {55-02},
      series = {Grad. Texts in Math.},
      address = {New York},
      isbn = {0-387-90336-4},
      publisher = {Springer-Verlag},
      zblnumber = {0406.55001},
      mrnumber = {0516508},
      mrreviewer = {Brayton Gray},
      title = {Elements of Homotopy Theory},
      year = {1978},
      pages = {xxi+744},
      }

Authors

Alexander Schmidt

Mathematisches Institut, Universität Heidelberg, Heidelberg, Germany

Jakob Stix

Institut für Mathematik, Goethe--Universität Frankfurt, Frankfurt am Main, Germany