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.

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.

Reviewer: Gerhard Pfister (Kaiserslautern)

##### 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 |