Semi-universal deformation spaces of some simple elliptic singularities
The Lie algebra \({\mathfrak g}={\mathfrak{sl}}(z, \mathbb C)\oplus {\mathfrak {sl}}(z, \mathbb C)\) is used to construct the semi-universal deformation of simple elliptic singularities of type \(\widetilde{D}_5\). The nilpotent variety of the Lie algebra above is \(\mathcal N=\{\binom{a\;b}{c-a}\;| \;a^2+bc=0\}\times \{\binom{d\;e}{f-d}\;| \;d^2+ef=0\}\). The surface singularity \((\mathcal N\cap S, 0)\) is simple elliptic of type \(\widetilde{D}_5\) for a generic slice \(S\) at \(0\). If the slice is defined by \(c=d+e\) and \(f=a+b\) the construction is as follows. Let \(\mathfrak{h}=\{\binom{a\;0}{0-a}\}\oplus\{\binom{d\;0}{0-d}\}\) and consider the adjoint quotient \(\chi:\mathfrak{g}\to\mathfrak{h}/W\;,\;\chi(\binom{a\;b}{c-a}, \binom{d\;e}{f-e})=(-a^2-bc, -d^2-ef)\). \(W\cong\mathbb Z/2\mathbb Z \oplus\mathbb Z/2\mathbb Z\) is the Weyl group of \(\mathfrak{g}\). For \((\alpha, \beta)\in \mathbb C^2\) let \(f_{(\alpha,\beta)}(\binom{a\;b}{c-a},\;\binom{d\;e}{f-e}) = (-a^2-bc-\alpha e, -d^2-ef-\beta b)\) and the slice \(S_{(\gamma, \delta, \varepsilon)}\) be defined by \(c=d+e+\gamma, f=a+b+\delta e+\varepsilon\). Let \(S:=\mathbb C^2\times \mathbb C^3\times \mathfrak{h}/W\) and \(X=\{(A, \alpha, \beta, \gamma, \delta, \varepsilon, \lambda, \mu)\in \mathfrak{g}\times S\;| \;f_{(\alpha,\beta)}(A)=(\lambda, \mu), A\in S_{(\gamma, \delta, \varepsilon)}\}\). Let \(o:=(0,0,0,0,0,0,0)\in S\) and \(q:=(0,o)\in X\) then the morphism \((X,q)\to (S,o)\) is a semi-universal deformation of the simple elliptic singularity defined by the slice \(c=d+e, f=a+b\).
scientific article; zbMATH DE number 5178530