Versal deformations of powers of volume forms

Abstract

Deformations of powers of volume forms of the kind $$ F\left( {x,\lambda } \right){{\left( {dx} \right)}^{\alpha }},x \in {{C}^{n}},F\left( {x,0} \right) = f\left( x \right),dx = d{{x}_{1}} \wedge ... \wedge d{{x}_{n}},\alpha \in C $$ are investigated. If f has an isolated singularity of multiplicity μ at the origin, then the form f(x)(dx) α has a μ-parameter versal deformation for almost every value of α. Exceptional values of α form a discrete set of negative rational numbers. Given f, the versal deformation can be obtained algorithmically. For non-exceptional values of α it is the same as the versal deformation of the germ of a function f.