Generic Torelli for semiquasihomogeneous singularities.

*(English)*Zbl 1025.32025
Libgober, Anatoly (ed.) et al., Trends in singularities. Basel: Birkhäuser. Trends in Mathematics. 115-140 (2002).

A natural local analog of the classical Torelli-type theorems on projective varieties is the attempt to characterize singularities by means of suitable invariants. The author of the present paper had introduced before an invariant \(BL(f)\) associated to an isolated hypersurface singularity (understood to be a germ of an analytic function \(f:(\mathbb{C}^{n+1},0) \to(\mathbb{C},0)\), singular at the origin only.) This object involves the so-called Brieskorn lattice of the singularity (a module closely related to the associated mixed Hodge structure), plus some extra topological information; it is invariant under right-equivalence. He has conjectured that \(BL(f)\) determines the right equivalence class of \(f\), and proved this in earlier work in a number of cases.

In the present paper, he presents a rather general and technically difficult result in the positive direction. Namely, if we restrict ourselves to semiquasihomogeneous hypersurface singularities which are “general enough”, then the conjecture holds. Let us be more precise. We say that a singularity \(f\) is semiquasihomogeneous if, for a suitably given system of weights \((w_0,\dots,w_n)\), with each \(w_i\) a positive rational number, and assigning to each monomial \(x_0^{w_0} \dots x_n^{w_n}\) the degree \(\sum^n_{i=0} a_iw_i\), then all the monomials in the expansion of \(f\) with nonzero coefficient have degree \(\geq 1\), and the sum \(f_0\) of those whose degree is \(=1\) define an isolated singularity. Then \(f\) itself has an isolated singularity, and the Milnor numbers of \(f_0\) and \(f\) agree. Concerning the genericity statement, it can be proved that there are moduli spaces (which are analytic varieties) \({\mathcal M}_{\text{sem}}\) and \(D_{BL}/G_\mathbb{Z}\) for right equivalence classes of isolated semiquasihomogeneous singularities \(f\) with a fixed Milnor number \(\mu\) and for certain equivalence classes of Brieskorn lattices respectively, and a “period” morphism \(\varphi:{\mathcal M}_{\text{sem}} \to D_{BL}/G_\mathbb{Z}\). In this paper Hertling shows that there is a proper nowhere dense analytic subset \({\mathcal M}'\subset {\mathcal M}_{\text{sem}}\) such that \(\varphi\), restricted to \({\mathcal M}_{\text{sem}} -{\mathcal M}'\) is injective. This easily implies that for singularities \(f\) and \(f'\), both corresponding to points of \({\mathcal M}_{\text{sem}}- {\mathcal M}'\), if \(BL(f)= BL(f')\) then \(f\) is right equivalent to \(f'\).

For the reader’s convenience, the paper includes a review of the basic definitions and background results (many obtained by Hertling himself in previous work).

For the entire collection see [Zbl 0997.00011].

In the present paper, he presents a rather general and technically difficult result in the positive direction. Namely, if we restrict ourselves to semiquasihomogeneous hypersurface singularities which are “general enough”, then the conjecture holds. Let us be more precise. We say that a singularity \(f\) is semiquasihomogeneous if, for a suitably given system of weights \((w_0,\dots,w_n)\), with each \(w_i\) a positive rational number, and assigning to each monomial \(x_0^{w_0} \dots x_n^{w_n}\) the degree \(\sum^n_{i=0} a_iw_i\), then all the monomials in the expansion of \(f\) with nonzero coefficient have degree \(\geq 1\), and the sum \(f_0\) of those whose degree is \(=1\) define an isolated singularity. Then \(f\) itself has an isolated singularity, and the Milnor numbers of \(f_0\) and \(f\) agree. Concerning the genericity statement, it can be proved that there are moduli spaces (which are analytic varieties) \({\mathcal M}_{\text{sem}}\) and \(D_{BL}/G_\mathbb{Z}\) for right equivalence classes of isolated semiquasihomogeneous singularities \(f\) with a fixed Milnor number \(\mu\) and for certain equivalence classes of Brieskorn lattices respectively, and a “period” morphism \(\varphi:{\mathcal M}_{\text{sem}} \to D_{BL}/G_\mathbb{Z}\). In this paper Hertling shows that there is a proper nowhere dense analytic subset \({\mathcal M}'\subset {\mathcal M}_{\text{sem}}\) such that \(\varphi\), restricted to \({\mathcal M}_{\text{sem}} -{\mathcal M}'\) is injective. This easily implies that for singularities \(f\) and \(f'\), both corresponding to points of \({\mathcal M}_{\text{sem}}- {\mathcal M}'\), if \(BL(f)= BL(f')\) then \(f\) is right equivalent to \(f'\).

For the reader’s convenience, the paper includes a review of the basic definitions and background results (many obtained by Hertling himself in previous work).

For the entire collection see [Zbl 0997.00011].

Reviewer: Augusto Nobile (Baton Rouge)

##### MSC:

32S25 | Complex surface and hypersurface singularities |

32S10 | Invariants of analytic local rings |

32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |

34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |

14B05 | Singularities in algebraic geometry |