Flattening and algebrisation

Michael McQuillan (University of Rome Tor Vergata)

Friday 6th December, 2024 13:00-14:00 Maths 311B

Abstract

Often natural moduli problems, e.g. foliated surfaces, come without an ample line bundle, so the
algebraisability of formal deformations, and the very existence of a moduli space requires a study
of the mermorphic functions on the aforesaid deformations, and the flattening (by blowing up) of the
resulting meromorphic maps. In  such a context the flattening theorem  of Raynaud  & Gruson, and derivatives thereof, is  close to  irrelevant since it  systematically uses  schemeness to  globally
extend local centres of  blowing up. This was  already well understood by  Hironaka in his proof  of
holomorphic flattening, and his ideas are the right ones. Nevertheless, the said ideas can be better
organised with a more systematic use of Grothendieck's universal flatifier, and, doing so, leads  to
a fully  functorial, and radically simpler,  proof provided the sheaf  of nilpotent  functions is
coherent-which is true for excellent formal schemes, but, unlike schemes or complex spaces, is false
in general.

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