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Lê's vanishing polyhedron for a family of mixed functions
Author(s) -
CisnerosMolina José Luis,
Menegon Aurélio
Publication year - 2019
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12299
Subject(s) - mathematics , homotopy , holomorphic function , polyhedron , combinatorics , fiber , spheres , type (biology) , gravitational singularity , polynomial , pure mathematics , mathematical analysis , physics , composite material , ecology , astronomy , biology , materials science
We study real analytic isolated singularities of type f : ( C n + m + 1 , 0 ) → ( C , 0 )with f ( z , w ) = g ( z ) + ∑ i = 1 m + 1h i ( w i , w ¯ i ) , where g is holomorphic and each h i is a mixed polynomial, with z = ( z 1 , ⋯ , z n ) and w = ( w 1 , ⋯ , w m + 1 ) . We construct a Lê's vanishing polyhedron for f , which describes the degeneration of its Milnor fiber F f to the singular fiber. Then we prove that F f is homotopy equivalent to the joinF g ∗ F h 1 ∗ ⋯ ∗ F h m + 1, where F g is the Milnor fiber of g and F h iis the Milnor fiber of h i . This implies that F f has the homotopy type of a bouquet of spheres S n + m . So we can define the Milnor number μ ( f ) as the number of spheres in that bouquet, as in the complex setting.