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Green-Lazarsfeld sets and resolvable quotients of Kähler groups. (Ensembles de Green-Lazarsfeld et quotients resolubles des groupes de Kähler.) (French) Zbl 1072.14512

From the introduction: Hodge theory allows one to obtain restrictions on the abelian or nilpotent quotients of \(\pi_1(X)\) when \(X\) is a compact Kähler manifold. In the abelian case, one has the classical parity of \(b_1(X)\). In the nilpotent case, one has, in particular, the quadraticity of the Mal’tsev Lie algebra of \(\pi_1(X)\) [see P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Invent. Math. 29, 245–274 (1975; Zbl 0312.55011)] and the fact that the descending central sequence of \(\pi_1(X)\) is equal up to torsion to that of \(\pi_1(Y)\) when \(Y\) is a smooth model of the image of \(X\) by its Albanese morphism [F. Campana, Ann. Sci. Éc. Norm. Supér., IV. Sér. 28, No. 3, 307–316 (1995; Zbl 0829.32006)].
In this paper we propose to study the solvable quotients of \(\pi_1(X)\), guided by the following two analogous (unproved) principles: (a) These quotients are equal up to torsion to those of \(\pi_1(Y)\). (b) They are determined by the Green-Lazarsfeld set \(\Sigma^1(X)\subset\operatorname{Hom}(\pi_1(X),\mathbb C^{*})\) of characters, whose first cohomology group is nonzero. This set \(\Sigma^1(X)\) will play in the solvable case a role analogous to that played in the nilpotent case by \[ \text{Ker}(\bigwedge^2H^1(X,\mathbb C)\to H^2(\mathbb C)), \] which determines the Mal’tsev Lie algebra of \(\pi_1(X)\) via the formality of \(X\).
We establish some simple consequences of these two principles: (1) \(\Sigma^1(X)\) is a finite union of translations of complex subtori of \(\text{Hom}(\pi_1(X),\mathbb C^{*})\) by torsion elements (section 1.3). To that end, we reduce to the projective case, known from C. Simpson [Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, No.3, 361–401 (1993; Zbl 0798.14005)], based on arguments of A. Beauville [in: Complex algebraic varieties (Bayreuth, 1990), 1–15, Lect. Notes Math. 1507 (1992; Zbl 0792.14006)]. (2) The solvable linear quotients of \(\pi_1(X)\) factor virtually (i.e., after finite étale covering) by the Stein reduction of the Albanese morphism of \(X\) (section 4.2). This partially establishes assertion (a) above. (3) The solvable quotients of \(\pi_1(X)\) are extensions of virtually nilpotent groups by torsion groups, under a condition of finiteness of the rank (§2.3). This condition means that no solvable Galois finite étale covering of \(X\) has a surjective morphism on a curve of genus \(g\geq 2\) (see section 2.7).
For projective \(X\), this result is due essentially to D. Arapura and M. Nori [Compos. Math. 116, No. 2, 173–188 (1999; Zbl 0971.14020)], who obtained it by methods of arithmetic geometry. In fact, their paper motivated the present work and provided numerous arguments for it (see section 2 and the appendix).
Our goal is to extend their results in the projective case to the Kähler case while avoiding the arithmetic methods as much as possible, although they remain crucial (because of the use of [C. Simpson, loc. cit.] in section 1).
In section 3, we compare the solvable quotients of \(\pi_1(X)\) and \(\pi_1(A)\), where \(A\) is the algebraic reduction of \(X\). In particular, if \(X\) is of algebraic dimension zero, these quotients are virtually abelian (section 3.2). This result agrees with the usual conjectures about these manifolds (section 3.4).
In section 4, we establish the factorization by the image of the Albanese map of the morphisms of \(\pi_1(X)\) in solvable linear groups (after possible étale finite covering). The technique used in this section, which rests on Deligne’s semisimplicity theorem, can in fact provide a simpler proof of all the above results. This will be shown in a subsequent work.
Some of the results in the present work are similar to those obtained by A. Brudnyi [Mich. Math. J. 46, No. 3, 489–514 (1999; Zbl 1082.32013)] by very different methods, in the case of certain representations with values in the group of upper triangular matrices.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
32J27 Compact Kähler manifolds: generalizations, classification
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