Subspaces of moduli spaces of rank one local systems. (English) Zbl 0798.14005

Simpson, Carlos

Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361-401 (1993).

Let \(X\) be a smooth projective variety. The author considers the moduli space \(M(X)\) of rank one local systems on \(X\). It is a group under tensor product. \(M(X)\) has three different structures of complex algebraic group: Betti, de Rham and Dolbeault: \(M_ B\), \(M_{DR}\), and \(M_{DOL}\). In particular \(M_ B = \operatorname{Hom} (\pi_ 1(X), \mathbb{C}^*)\). \(M_{DR}\) is the moduli space of pairs \((L,\nabla)\), where \(L\) is a line bundle on \(X\) and \(\nabla\) is an integrable algebraic connection on \(L\). \(M_{DOL} (X)\) is the moduli space of rank one Higgs bundles with first Chern class vanishing in the cohomology with rational coefficients. A rank one Higgs bundle is \((E, \varphi)\) where \(E\) is a line bundle and \(\varphi \in H^ 0 (X, \Omega^ 1_ X)\). A subgroup which is algebraic for all three structures is called a triple torus. Main theorem:
Suppose that \(S \subset M\) is a closed irreducible real analytic subset. Suppose that one of the following sets of hypotheses holds:
(a) \(S_ B\) is complex analytic and \(S_{DOL}\) is preserved by \(\mathbb{C}^*\) action.
(b) \(S_ B\) is complex analytic and \(S_{DOL}\) is complex algebraic.
(c) \(S_ B\) and \(S_{DR}\) are complex algebraic.
Then \(S\) is a translate of a triple tori by torsion points. Here \(S_ B\), \(S_{DR}\) and \(S_{DOL}\) denote the corresponding subsets of \(M_ B\), \(M_{DR}\), and \(M_{DOL}\).
The proof rests on a result from transcendental number theory. Thus any closed subspace \(S\) of \(M(X)\) which is defined in a natural way, for example, by looking at cohomologies and related constructions, is a finite union of translates of triple tori by torsion points.

Reviewer: V.K.Vedernikov (Moskva)

MSC:

14D20	Algebraic moduli problems, moduli of vector bundles
14L05	Formal groups, \(p\)-divisible groups

Keywords:

complex algebraic group structure of moduli space; rank one Higgs bundles; triple torus

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