On \(c_ 1=0\) resolution of quotient singularity.

*(English)*Zbl 0856.14005The purpose of this paper is to study minimal resolutions of quotient singularities with trivial canonical sheaf, more precisely the minimal resolution of the 3-dimensional quotient singularity by the isocahedral group and some higher dimensional abelian quotients.

Let \(\pi:V \to X\) the canonical projection from the vector space \(V=\mathbb{C}^n\) to the quotient space \(X=V/G\), where \(G\) is a finite subgroup of \(SL_n(\mathbb{C})\). The author wants to compare the Euler number \(\chi(X)\) of some minimal resolution of \(X\) with the “orbifold Euler characteristic” \(\chi(V,G)\). In a first part the author considers a double cover \(p:X\to Y\) of a smooth variety defined by some \(s \in \Gamma (Y, {\mathcal K}^2_Y)\), where \({\mathcal K}_Y\) is the canonical line bundle on \(Y\). Let \(B\) the ramification locus of \(p\) and assume \(\text{Sing} (B)=\bigcup^k_{j=1} C_j\), where each \(C_j\) is a non-singular codimension 2 subvariety of \(Y\). Then \(X\) is a normal variety with trivial canonical sheaf and \(\text{Sing} (X)=p^{-1} (\text{Sing} (B))\). – By induction we can construct a modification \(\widetilde \sigma : \widetilde Y\to Y\), as the composition of blow-up of a curve \(C_j\), such that the double cover \(\widetilde X\) of \(\widetilde Y\) ramified at the strict transform of \(B\) on \(\widetilde Y\) gives an explicit resolution \(\widetilde \tau: \widetilde X \to X\) with trivial canonical bundle.

Then the author studies the space \(X=\mathbb{C}^3/I\) defined as the quotient 3-space by the icosahedral group \(I\). If we denote by \(J\) the subgroup of \(GL_3 (\mathbb{C})\) generated by \(I\) and \(-\text{id}\), then we get a double cover \(p : X \to Y= \mathbb{C}^3/J\). The quotient 3-space \(Y\) is smooth and the singular set of the branch locus \(B\) is the union of three smooth curves. Then we can apply the previous procedure and construct a resolution \(\widetilde X\) of \(X\) and we can calculate its Euler number \(\chi (\widetilde X)\) and we get the equality \(\chi (\widetilde X) = \chi (\mathbb{C}^3,I) = 5\).

In the same way we can consider the quotient \(n\)-space \(X=\mathbb{C}^n/SD\), where \(SD=D \cap SL_n (\mathbb{C}) \simeq (\mathbb{Z}/2)^{n-1}\) and \(D\) is the subgroup of \(GL_n (\mathbb{C})\) consisting of all the diagonal order 2 elements, as a double cover of the smooth variety \(Y=\mathbb{C}^n/D\). We get a minimal resolution \(\widetilde X\) of \(X\) with trivial canonical bundle and we have the equality \(\chi (\widetilde X) = \chi (\mathbb{C}^n,SD)=2^{n-1}\).

Let \(\pi:V \to X\) the canonical projection from the vector space \(V=\mathbb{C}^n\) to the quotient space \(X=V/G\), where \(G\) is a finite subgroup of \(SL_n(\mathbb{C})\). The author wants to compare the Euler number \(\chi(X)\) of some minimal resolution of \(X\) with the “orbifold Euler characteristic” \(\chi(V,G)\). In a first part the author considers a double cover \(p:X\to Y\) of a smooth variety defined by some \(s \in \Gamma (Y, {\mathcal K}^2_Y)\), where \({\mathcal K}_Y\) is the canonical line bundle on \(Y\). Let \(B\) the ramification locus of \(p\) and assume \(\text{Sing} (B)=\bigcup^k_{j=1} C_j\), where each \(C_j\) is a non-singular codimension 2 subvariety of \(Y\). Then \(X\) is a normal variety with trivial canonical sheaf and \(\text{Sing} (X)=p^{-1} (\text{Sing} (B))\). – By induction we can construct a modification \(\widetilde \sigma : \widetilde Y\to Y\), as the composition of blow-up of a curve \(C_j\), such that the double cover \(\widetilde X\) of \(\widetilde Y\) ramified at the strict transform of \(B\) on \(\widetilde Y\) gives an explicit resolution \(\widetilde \tau: \widetilde X \to X\) with trivial canonical bundle.

Then the author studies the space \(X=\mathbb{C}^3/I\) defined as the quotient 3-space by the icosahedral group \(I\). If we denote by \(J\) the subgroup of \(GL_3 (\mathbb{C})\) generated by \(I\) and \(-\text{id}\), then we get a double cover \(p : X \to Y= \mathbb{C}^3/J\). The quotient 3-space \(Y\) is smooth and the singular set of the branch locus \(B\) is the union of three smooth curves. Then we can apply the previous procedure and construct a resolution \(\widetilde X\) of \(X\) and we can calculate its Euler number \(\chi (\widetilde X)\) and we get the equality \(\chi (\widetilde X) = \chi (\mathbb{C}^3,I) = 5\).

In the same way we can consider the quotient \(n\)-space \(X=\mathbb{C}^n/SD\), where \(SD=D \cap SL_n (\mathbb{C}) \simeq (\mathbb{Z}/2)^{n-1}\) and \(D\) is the subgroup of \(GL_n (\mathbb{C})\) consisting of all the diagonal order 2 elements, as a double cover of the smooth variety \(Y=\mathbb{C}^n/D\). We get a minimal resolution \(\widetilde X\) of \(X\) with trivial canonical bundle and we have the equality \(\chi (\widetilde X) = \chi (\mathbb{C}^n,SD)=2^{n-1}\).

Reviewer: M.Vaquie (Paris)