\(ADE\) bundles over surfaces with \(ADE\) singularities (Q2929641)

The article studies relations between ADE-Lie theory and ADE-singularities of surfaces.NEWLINENEWLINELet \(X\) denote a compact complex surface with a rational double point and \(\pi :Y \to X\) the minimal resolution of. If \(C_1 , \dots , C_n\) in \(Y\) are the irreducible components of the exceptional locus, then the dual graph of the exceptional divisor \(\sum_{i=1}^n C_i\) is a Dynkin diagram of one of the types A, D, E.NEWLINENEWLINEFor the integer homology of \(Y\), there is a natural decomposition \(H^2(Y, {\mathbb{Z}} )= H^2(X, {\mathbb{Z}} ) \oplus \Lambda \), where \(\Lambda = \{ \sum _i a_i[C_i] | a_i \in {\mathbb{Z}} \} \). The subset \(\Phi := \{ \alpha \in \Lambda | \alpha^2 =-2 \}\) is an ADE root system of a simple Lie-algebra \(\mathbf g \). Its associated Lie algebra bundle over \(Y\) is defined as NEWLINE\[NEWLINE {\mathcal E}_0^{\mathbf g}:= {\mathcal O_Y}^n \oplus \{ \bigoplus_{\alpha \in \Phi}{\mathcal O_Y} (\alpha ) \} .NEWLINE\]NEWLINENEWLINENEWLINEThis bundle does not descend to the original surface \(X\). The authors show, if \(p_g(X)=0\) then \( {\mathcal E}_0^{\mathbf g}\) has a deformation to a bundle which can descend to \(X\).NEWLINENEWLINETheir result generalizes the work of Friedman-Morgan for \(E_n\)-bundles over del Pezzo surfaces [\textit{R. Friedman} and \textit{J. W. Morgan}, Contemp. Math. 312, 101--115 (2002; Zbl 1080.14533)].NEWLINENEWLINEFurthermore, the authors describe the minuscule representation bundles of these Lie algebra bundles in terms of configurations of (reducible) \((-1)\)-curves in \(Y\).