# zbMATH — the first resource for mathematics

Frobenius manifolds and moduli spaces for singularities. (English) Zbl 1023.14018
Cambridge Tracts in Mathematics. 151. Cambridge: Cambridge University Press. ix, 270 p. £45.00; \$ 60.00 (2002).
Frobenius manifolds are complex manifolds with an additional structure on the holomorphic tangent bundle: a multiplication and a metric which are compatible in a canonical way. They originate from physics and play a role in quantum cohomology and mirror symmetry. The book under review shows a beautiful application to singularity theory, the construction of global moduli spaces for isolated hypersurface singularities. The basis of the book is the author’s habilitation.
Part I of the book under review is devoted to the local structure of $$F$$-manifolds. They are closely related to singularity theory and symplectic geometry. An $$F$$-manifold is a complex manifold $$X$$ such that each holomorphic tangent space $$T_{X,x}$$ is a commutative and associative algebra with unit element, and the multiplication varies in a specific way with the point $$x$$. Studying $$F$$-manifolds, one is led to discriminants, a classical subject of singularity theory and to Lagrange maps and their singularities.
A Frobenius manifold is an $$F$$-manifold, together with a flat metric $$g$$ and an Euler field $$E$$, such that $$\text{Lie}_e(g) = 0$$, $$e$$ the global unit field of the $$F$$-manifold, $$\text{Lie}_E(g) = D \cdot g$$ for some $$D \in \mathbb{C}$$.
Part II of the book is devoted to the construction of Frobenius manifolds in singularity theory. The base space of a semi-universal unfolding of an isolated hypersurface singularity can be equipped with the structure of a Frobenius manifold (results of K. Saito and M. Saito). The construction involves the Gauß-Manin connection and polarised mixed Hodge structures. The highlight is the following construction of the global moduli space for isolated hypersurface singularities.
Let $$R = \{\varphi : (\mathbb{C}^n,0) \to (\mathbb{C}^k,0)$$, analytic coordinate changes$$\}$$ and $$J_k(R)$$ be the algebraic group of $$k$$-jets of coordinate changes on $$\mathfrak{m}^2/\mathfrak{m}^{k+1}$$, $$\mathfrak{m}$$ the maximal ideal in $$\mathcal{O}_{\mathbb{C}^n,0}$$. Fix integers $$\mu$$ and $$k \geq \mu+1$$ and $$f \in \mathfrak{m}^2$$. Let $$C(k,f) \subseteq \{J_k(g) \mid \mu(g) = \mu\}$$ be the topological component containing $$f$$. Then $$J_k(R)$$ acts on $$C(k,f)$$; the quotient $$C(k,f) / J_k(R)$$ is an analytic geometric quotient. The germ at $$[J_k(f)]$$ is isomorphic to $$(S_\mu,0)/\text{Aut}\bigl((M,0),o,e,E\bigr)$$. Here $$\bigl((M,0), o,e,E\bigr)$$ is the base space of a semi-universal unfolding of $$f$$ with its structure as a germ of an $$F$$-manifold (with Euler field $$E$$) and $$(S_\mu,0) \subset (M,0)$$ is the $$\mu$$-constant stratum.
The book gives a nice introduction to the theory of Frobenius manifolds and shows how they can be used in singularity theory. It can be used as a basis for researchers and graduate students to work in this area.

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 14J17 Singularities of surfaces or higher-dimensional varieties 14B05 Singularities in algebraic geometry