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

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.

### MSC:

 14D20 Algebraic moduli problems, moduli of vector bundles 14L05 Formal groups, $$p$$-divisible groups
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### References:

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