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On the uniformization of compact Kähler orbifolds. (English) Zbl 1318.32029

The article under review studies the uniformization problem for compact Kähler orbifolds. In the first section, the author proves certain results related to the stacky version of the Serre problem (i.e., the problem of characterizing the finitely presentable groups which occur as the fundamental group of a compact Kähler orbifold) and the orbifold version of the Shafarevich conjecture.
In the second section, the author studies the Shafarevich conjecture for compact Kähler orbifolds whose fundamental group is linear. Note that in the compact Kähler manifold case, this linear Shafarevich conjecture was proved in [F. Campana, F. Claudon and P. Eyssidieux, Compos. Math. 151, No. 2, 351–376 (2015; Zbl 1432.32029); P. Eyssidieux, L. Katzarkov, T. Pantev and M. Ramachandran, Ann. Math. (2) 176, No. 3, 1545–1581 (2012; Zbl 1273.32015)].
In the third section, the author studies the singular case. Note that the universal covering space of a nodal rational curve is not holomorphically convex. The author observes that the uniformization problem in the singular case might be related to non-abelian mixed Hodge theory. The article gives some evidence in this direction.
Reviewer: Junyan Cao (Paris)

MSC:

32Q30 Uniformization of complex manifolds
32J27 Compact Kähler manifolds: generalizations, classification
32E05 Holomorphically convex complex spaces, reduction theory
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