## The Milnor fiber conjecture of Neumann and ...

Document type :

Partie d'ouvrage

DOI :

Title :

The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof

Author(s) :

Cueto, Maria Angelica [Auteur]

Mathematics Department, The Ohio State University

Popescu-Pampu, Patrick [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]

Stepanov, Dmitry [Auteur]

Moscow Institute of Physics and Technology [Moscow] [MIPT]

Mathematics Department, The Ohio State University

Popescu-Pampu, Patrick [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]

Stepanov, Dmitry [Auteur]

Moscow Institute of Physics and Technology [Moscow] [MIPT]

Scientific editor(s) :

Athanase Papadopoulos

Book title :

Essays in Geometry

Publisher :

EMS Press

Publication date :

2023-08-08

ISBN :

978-3-98547-024-2

English keyword(s) :

Complete intersection singularities

integral homology spheres

Kato-Nakayama spaces

local tropicalization

log geometry

Milnor fibers

Newton non-degeneracy

real oriented blowups

rounding

Seifert fibrations

surface singularities

splice type singularities

toric geometry

toroidal varieties

tropical geometry

integral homology spheres

Kato-Nakayama spaces

local tropicalization

log geometry

Milnor fibers

Newton non-degeneracy

real oriented blowups

rounding

Seifert fibrations

surface singularities

splice type singularities

toric geometry

toroidal varieties

tropical geometry

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. ...

Show more >Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. They are determined, up to a form of equisingularity, by decorated trees called splice diagrams. In 2005, Neumann and Wahl formulated their Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a special kind of decomposition into pieces of the Milnor fibers of the associated singularities. These pieces are constructed from the Milnor fibers of the splice type singularities determined by the subdiagrams on both sides of the chosen edge. In this paper we give an overview of this conjecture and a detailed outline of its proof, based on techniques from tropical geometry and log geometry in the sense of Fontaine and Illusie. The crucial log geometric ingredient is the operation of rounding of a complex logarithmic space introduced in 1999 by Kato and Nakayama. It is a functorial generalization of the operation of real oriented blowup. The use of the latter to study Milnor fibrations was pioneered by A’Campo in 1975.Show less >

Show more >Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. They are determined, up to a form of equisingularity, by decorated trees called splice diagrams. In 2005, Neumann and Wahl formulated their Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a special kind of decomposition into pieces of the Milnor fibers of the associated singularities. These pieces are constructed from the Milnor fibers of the splice type singularities determined by the subdiagrams on both sides of the chosen edge. In this paper we give an overview of this conjecture and a detailed outline of its proof, based on techniques from tropical geometry and log geometry in the sense of Fontaine and Illusie. The crucial log geometric ingredient is the operation of rounding of a complex logarithmic space introduced in 1999 by Kato and Nakayama. It is a functorial generalization of the operation of real oriented blowup. The use of the latter to study Milnor fibrations was pioneered by A’Campo in 1975.Show less >

Language :

Anglais

Audience :

Internationale

Popular science :

Non

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