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Higgs bundles and local systems. (English) Zbl 0814.32003
From the introduction: “Let \(X\) be a compact Kähler manifold. We study a correspondence between representations of the fundamental group of \(X\), and certain holomorphic objects on \(X\). A Higgs bundle is a pair consisting of a holomorphic vector bundle \(E\), and a holomorphic map \(\theta : E \to E \otimes \Omega^ 1_ X\) such that \(\theta \wedge \theta = 0\). There is a condition of a stability analogous to the condition for vector bundles, but with reference only to subsheaves preserved by the map \(\theta\). There is a one-to-one correspondence between irreducible representations of \(\pi_ 1 (X)\), and stable Higgs bundles with vanishing Chern classes. This theorem is a result of several recent extension of the work of M. S. N. Narasimhan and C. S. Seshadri. The purpose of this paper is to discuss this correspondence in detail, to obtain some further properties, and to give some applications.
The correspondence between Higgs bundles and local systems can be viewed as a Hodge theorem for nonabelian cohomology. To understand this, let us first look at abelian cohomology: \(H^ 1 (X, \mathbb{C})\) can be thought of as the space of homomorphisms from \(\pi_ 1 (X)\) into \(\mathbb{C}\), or equivalently as the space of closed one-forms modulo exact one-forms. But since \(X\) is a compact Kähler manifold, the Hodge theorem gives a decomposition \[ H^ 1 (X, \mathbb{C}) = H^ 1 (X,{\mathcal O}_ X) \oplus H^ 0 (X, \Omega^ 1_ X). \] In other words, a cohomology class can be thought of as a pair \((e, \xi)\) with \(e \in H^ 1 (X,{\mathcal O}_ X)\) and \(\xi\) a holomorphic one-form. The correspondence between Higgs bundles and local systems is analogous. If \(\pi_ 1 (X)\) acts trivially on \(\text{Gl} (n, \mathbb{C})\) then the nonabelian cohomology set \(H^ 1 (\pi_ 1 (X), \text{Gl} (n,\mathbb{C}))\) is the set of representation \(\pi_ 1(X) \to \text{Gl} (n,\mathbb{C})\), up to conjugacy. Equivalently it is the set of isomorphism classes of \(C^ \infty\) vector bundles with flat connections. The theorem stated above gives a correspondence between the set of semisimple representations and the set of pairs \((E, \theta)\) where \(E\) is a holomorphic bundle (in other words, an element of \(H^ 1 (X, \text{Gl} (n, {\mathcal O}_ X))\) and \(\theta\) is an endomorphism valued one-form, subject to various additional conditions.
There is a natural action of \(\mathbb{C}^*\) on the set of Higgs bundles. A nonzero complex number \(t\) sends \((E, \theta)\) to \((E,t \theta)\). This preserves the conditions of stability and vanishing of Chern classes, so it gives an action on the space of semisimple representations. This \(\mathbb{C}^*\) action should be thought of as the Hodge structure on the semisimplified nonabelian cohomology.”
Contents of the paper: 0. Introduction. 1. Non-abelian Hodge theory. 2. Further properties. 3. Extensions. 4. Variations of Hodge structure. 5. The Hodge structure on the fundamental group. 6. Tannakian considerations.

MSC:
32G13 Complex-analytic moduli problems
14D07 Variation of Hodge structures (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations
32L05 Holomorphic bundles and generalizations
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