By Claus Hertling

ISBN-10: 0521812968

ISBN-13: 9780521812962

For these operating in singularity idea or different components of complicated geometry, this quantity will open the door to the examine of Frobenius manifolds. within the first half Hertling explains the idea of manifolds with a multiplication at the tangent package deal. He then provides a simplified clarification of the position of Frobenius manifolds in singularity conception in addition to the entire useful instruments and a number of other functions. Readers will reap the benefits of this cautious and sound examine of the elemental buildings and ends up in this intriguing department of arithmetic.

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Let (M, p) be a germ of a massive F-manifold. 14) is a free O M (r ) , p(r ) -module of rank n. It is an O M (r ) , p(r ) -algebra because of Liee (◦) = 0 · ◦. The functions a(X ) for X ∈ G p(r ) are invariant with respect to e and induce holomorphic functions on L (r ) . One obtains a map a(r ) : G p(r ) → π∗(r ) O L (r ) p (r ) . 17 The map a(r ) is an isomorphism of O M (r ) , p(r ) -algebras. Proof: The isomorphism a : T M, p → (π∗ O L ) p maps the e-invariant vector fields in (M, p) to the e-invariant functions in (π∗ O L ) p .

Such a function F will be fixed. It can be considered as a multivalued function on M − K ; the 1–graph of this multivalued function is L − π −1 (K ). The Lyashko–Looijenga map = ( 1 , . . , n ) : M → Cn of L ⊂ T ∗ M and F is defined as follows: for q ∈ M − K , the roots of the unitary polynomial n n−i are the values of F on π −1 (q). It extends to a holomorphic z n + i=1 i (q)z map on M because F is holomorphic on L reg and continuous on L. ) ) , . . , (red ): M → The reduced Lyashko–Looijenga map (red ) = ( (red n 2 n−1 ∗ of L ⊂ T M and F is defined as follows: for q ∈ M − K , the roots C (red ) n (q)z n−i are the values of F on of the unitary polynomial z n + i=2 i 1 1 −1 π (q), shifted by their centre − n 1 (q) = n λ∈π −1 (q) F(λ).

The last statement follows with (a). 4). The e-invariant hypersurfaces B and K project to hypersurfaces in M (r ) , which are called the restricted bifurcation diagram B(r ) and the restricted caustic K(r ) . 3 the restricted Lagrange map was defined as the germ at p (r ) ∈ M (r ) of a Lagrange map L (r ) → T ∗ M (r ) → M (r ) . 12 the notion of a reduced Lyashko–Looijenga map is welldefined for the restricted Lagrange map (independently of the identification of the Lagrange fibration with the cotangent bundle of M (r ) ).

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Frobenius Manifolds and Moduli Spaces for Singularities by Claus Hertling


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