The Monodromy Groups of Isolated Singularities of Complete Intersections
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1293 Wolfgang Ebeling
The Monodromy Groups of Isolated Singularities of Complete Intersections
Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches Institut der Universitat und Max-Planck-Institut fUr Mathematik, Bonn - vol. 12 F. Hirzebruch Adviser:
1293 Wolfgang Ebeling
The Monodromy Groups of Isolated Singularities of Complete Intersections
Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Wolfgang Ebeling Mathematisches Institut der Universitat Wegelerstr. 10,5300 Bonn 1, Federal Republic of Germany
Mathematics Subject Classification (1980): 14B05, 14D05, 14M 10, 32B30, 57R45; 10C30, 10E45, 20H 15, 51 F 15 ISBN 3-540-18686-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18686-7 Springer-Verlag New York Berlin Heidelberg
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For Bettina, Bastian, and Mirja
INTRODUCTION For the analysis of hyper surface singularities and their deformations the Milnor lattice and the invariants associated with it play an 1\
important role. The middle homology group of a Milnor fibre of the singularity is a free abelian group, which is endowed with a bilinear form, the intersection form. This bilinear form is symmetric or skewsymmetric, if the dimension is even or odd respectively. This group with this additional structure is called the Milnor lattice monodromy group
r
H. The
of the singularity is a subgroup of the automorp.hism
group of this lattice. It is generated by reflections, respectively symplectic transvections, corresponding to certain elements of the Milnor lattice, the vanishing cyCles, which are defined in a geometric way. These form the set
c H
of vanishing cycles. The monodromy group is al
ready generated by the respective automorphisms corresponding to the elements of certain geometrically distinguished bases of vanishing cycles. The set of Dynkin diagrams (or intersection diagrams) corresponding to such bases yields another invariant. A survey of the relations between these invariants and their importance for the deformation theory of the singularities in the hypersurface case is given by E. Brieskorn in his expository article [Brieskorn
3]. By stabilizing we can restrict ourselves to the symmetric case in
the hypersurface situation. For the simple hypersurface singularitie
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