Linear Shafarevich conjecture. (English) Zbl 1273.32015

This article consists in the strongest to date result in uniformization theory: it indeed settles the linear case of the Shafarevich conjecture which asserts that the universal cover of a smooth projective variety should be holomorphically convex. The main theorem can be sumed up in the following form.
Theorem. Let \(X\) be a smooth projective variety and \(n\geq1\) be an integer. If \(H_n\) denotes the intersection of the kernels of representations \(\pi_1(X)\longrightarrow \mathrm{GL}_n(A)\) where \(A\) is any \(\mathbb{C}\)-algebra of finite type, then the covering corresponding to \(H_n\) is holomorphically convex.
The reductive case (i.e., when only reductive representations are considered) was the object of a preceeding work of the first author [Invent. Math. 156, No. 3, 503–564 (2004; Zbl 1064.32007)]. The present paper has then to deal with general representations, and to do so the authors introduce new tools in nonabelian Hodge theory. The main assertions can be described in this way:
A strictness result (Proposition 3.6) which in some sense states that the relevant part of the unipotent completion is reduced to the abelian one (and the proof is reminiscent of the arguments given by the second author [J. Differ. Geom. 45, No. 2, 336–348 (1997; Zbl 0876.14008)]). From the mixed Hodge theoretical view point, Prop. 3.6 enables the authors to consider only variation of mixed Hodge structure (VMHS for short) whose weight filtration has length two.
The rationality lemma (Theorem 4.9) according to which some weight one sub-Hodge structure naturally defined in terms of \(\mathbb{C}\)-VHS is actually defined over \(\mathbb{Q}\).
In the end, the proof of both of these results resorts to the VMHS constructed by the first author and C. Simpson in [J. Eur. Math. Soc. (JEMS) 13, No. 6, 1769–1798 (2011; Zbl 1246.14018)].
The proof of the linear Shafarevich conjecture goes then roughly in the following way: thanks to the strictness argument (1), we only have to deal with VMHS of lenght 2 and the authors show that there is a prefered one (in the situation they consider). This VMHS comes equipped with a period domain which is an affine bundle over a Griffiths domain. Using this and the Shafarevich map (from the reductive case in [Zbl 1064.32007]), they produce a map from some covering of \(X\) to the product of the period domain and of the Shafarevich variety whose every connected component of a fiber is compact. And these subvarieties are actually the maximal compact subvarieties of the covering considered; this is a consequence of the rationality Lemma (2). The Stein factorization of this map produces then the Cartan-Remmert reduction: the base of the former is an affine bundle over a Stein space and then is Stein as well.


32E05 Holomorphically convex complex spaces, reduction theory
32Q30 Uniformization of complex manifolds
32G20 Period matrices, variation of Hodge structure; degenerations
14F35 Homotopy theory and fundamental groups in algebraic geometry
14D07 Variation of Hodge structures (algebro-geometric aspects)
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