By Claus Hertling
Cambridge Tracts in Mathematics (No. 151)
Publisher: Cambridge University Press
Print Publication Year: 2002
Online Publication Date:September 2009
Online ISBN:9780511543104
Hardback ISBN:9780521812962
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511543104.005
Subjects: Geometry and topology
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Discriminants play a central role in singularity theory. Usually they have a rich geometry and say a lot about the mappings or other objects from which they are derived. The discriminant D of a massive F-manifold M with a generating function (cf. Definition 3. 18) is an excellent model case of such discriminants, having many typical properties.
Together with the unit field it determines the whole F-manifold in a nice geometric way. This is discussed in section 4.1 (cf. Corollary 4.6). In section 4.3 results from singularity theory are adapted to show that the discriminant and also the bifurcation diagram are free divisors under certain hypotheses.
The classification of germs of 2-dimensional massive F-manifolds is nice and is carried out in section 4.2. Already for 3-dimensional germs it is vast (cf. section 5.5). In section 4.4 the Lyashko–Looijenga map is used to prove that the automorphism group of a germ of a massive F-manifold is finite. There also the notions modality and μ-constant stratum from singularity theory are adapted to F-manifolds. In section 4.5 the relation between analytic spectrum and multiplication is generalized. This allows F-manifolds to be found in natural geometric situations (e.g. hypersurface and boundary singularities) and to be written down in an economic way (e.g. in 5.22, 5.27, 5.30, 5.32).
pp. i-iv
pp. v-vii
pp. viii-x
Part 1 - Multiplication on the tangent bundle: Read PDF
pp. 1-2
1 - Introduction to part 1: Read PDF
pp. 3-8
2 - Definition and first properties of F-manifolds: Read PDF
pp. 9-22
3 - Massive F-manifolds and Lagrange maps: Read PDF
pp. 23-39
4 - Discriminants and modality of F-manifolds: Read PDF
pp. 40-60
5 - Singularities and Coxeter groups: Read PDF
pp. 61-96
Part 2 - Frobenius manifolds, Gauß–Manin connections, and moduli spaces for hypersurface singularities: Read PDF
pp. 97-98
6 - Introduction to part 2: Read PDF
pp. 99-108
7 - Connections over the punctured plane: Read PDF
pp. 109-130
8 - Meromorphic connections: Read PDF
pp. 131-144
9 - Frobenius manifolds and second structure connections: Read PDF
pp. 145-164
10 - Gauß–Manin connections for hypersurface singularities: Read PDF
pp. 165-194
11 - Frobenius manifolds for hypersurface singularities: Read PDF
pp. 195-217
12 - μ-constant stratum: Read PDF
pp. 218-229
13 - Moduli spaces for singularities: Read PDF
pp. 230-247
14 - Variance of the spectral numbers: Read PDF
pp. 248-259
pp. 260-268
pp. 269-270