Some structure theorems for semi-simple representations of \(\pi_ 1\) of algebraic manifolds
Let \(X\) be a compact algebraic variety with a fixed ample line bundle \(H\), and \(x \in X\) a base point. The object of this paper is an investigation of semi-simple representations of the fundamental group \(\rho \in \Hom (\pi_1 (X,x), G)^{ss}\), where \(G \subset SL_r (\mathbb{C})\) is a semisimple Lie-group. The quotient space \(\Hom (\pi_1 (X,x), G)^{ss}/G\) is called the Betti space. A representation \(\rho\) is called rigid if \(\rho\) is an isolated point in the Betti space; \(\rho\) is called étale-reducible, if there exists a finite étale covering \(\varphi : Z \to X\), so that the pull back of \(\rho\) to the universal covering of \(X\) splits as a nontrivial direct sum of simple- representations; \(\rho\) is called fibration-reducible, if there is a surjective morphism \(f : X \to B\) with connected fibres, so that the restriction representation of \(\rho\) to a generic fibre splits as a nontrivial direct sum of simple representations, and \(\rho\) is called unitary fibration-reducible, if each component in the direct sum is a unitary representation. -- The following result are obtained: Theorem 1: Let \(G = SL_r (\mathbb{C})\) and the Higgs bundle corresponding to \(\rho\) is not nilpotent. Then: (1) either \(\rho\) is étale-reducible, if \(c_1(I) H^{\dim H - 1} = 0\), where \(I\) is the image sheaf corresponding to the Higgs structure; or (2) there exists a finite étale covering \(\varphi : Z \to X\) so that the pull back \(\varphi^* \rho\) is unitary fibration-reducible, and the base space is an algebraic curve, if \(c_1 (I) H^{\dim H - 1} > 0\), and rank \(I = 1\). If \(r = 2\), then \(\rho\) in (2) is unitary fibration- reducible. Theorem 2: If \(G = SL_2 (\mathbb{C})\), then there are only three possibilities: either \(\rho\) is rigid, or \(\rho\) is étale-rigid, in fact, a double covering, or \(\rho\) is unitary fibration-reducible and the base space is an algebraic curve. Theorem 3: If \(G = SL_3 (\mathbb{C})\), there are only three possibilities: either \(\rho\) is rigid, or \(\rho\) is étale-reducible, or \(\rho\) is fibration-reducible, and the base space is an algebraic curve or a surface.
representations of the fundamental group
scientific article; zbMATH DE number 537257
scientific article