Orbifold Kähler groups and the Shafarevich conjecture for Hirzebruch's covering surfaces with equal weights (Q1642924)

The paper studies some aspects of the theory of Kähler groups (fundamental groups of compact Kähler manifolds) and in particular the currently widely disbelieved conjecture of Shafarevich that the universal covering space of a complex projective manifold should be holomorphically convex. A counterexample would have to have non-linear fundamental group, but that can happen as was shown by \textit{D. Toledo} [Publ. Math., Inst. Hautes Étud. Sci. 77, 103--119 (1993; Zbl 0818.14009)]. Rather than study fundamental groups, the author chooses to study orbifold Kähler groups, which are more easily constructed and controlled. It is not known whether orbifold Kähler groups are Kähler groups in general: it is true for residually finite groups, but the fundamental group in Toledo's example is not residually finite either. Still, the two classes are sufficiently close to make the investigation interesting. Orbifolds are here treated as stacks, which proves to be a convenient approach. An orbifold \({\mathcal X}\) is said to be developable if its universal cover in the stack sense is a manifold. This amounts to saying that the local inertia group \(I_x\) injects into \(\pi_1({\mathcal X})\) for every orbifold point of \({\mathcal X}\). It is said to be uniformisable if \(I_x\) injects into \(\pi_1^{\text{ét}}({\mathcal X})\). One of the results here is that developable does not imply uniformisable, even in the case of smooth orbifold locus, because Toledo's example is developable but not uniformisable. It is also shown that every orbifold Kähler group arises as the fundamental group of a developable orbifold. The second part of the paper examines the case of the Hirzebruch covering surfaces \({\mathcal M}_N({\mathcal A})\), \(N\)-fold covers of \({\mathbb P}^2\) branched along certain line arrangements \({\mathcal A}\subset{\mathbb P}^2\) as described in [\textit{G. Barthel} et al., Geradenkonfigurationen und algebraische Flächen. Braunschweig/Wiesbaden: Friedr. Vieweg \& Sohn (1987; Zbl 0645.14016)]. The author takes the opportunity to explain when \(\pi_1({\mathcal M}_N({\mathcal A}))\) (misprinted here as \(\pi_1({\mathcal M}_2({\mathcal A}))\) is finite: this happens if and only if \({\mathcal A}\) has only double points, or double and triple points if \(N=2\), and is asserted in [\textit{L.-C. Lefèvre}, Manuscr. Math. 152, No. 3--4, 381--397 (2017; Zbl 1386.14086)] on the basis of a letter from the author of this paper. Then he turns to the Shafarevich conjecture, and shows that it does hold for \({\mathcal M}_N({\mathcal A})\), assuming that the lines in \({\mathcal A}\) do not all pass through a single point. The final section contains some open questions, with some partial answers. One is whether \(\pi_1({\mathcal M}_N({\mathcal A}))\) is always linear, or always residually finite. The author observes that it is not a in general a product of curve orbifold fundamental groups (for \(N=5\), the rational cohomological dimension is wrong). He asks whether the universal cover of \({\mathcal M}_N({\mathcal A})\) is holomorphically convex, expecting that it is and that it should not be very hard to prove it. Also in the final section one finds a group theoretical consequence (or version) of the Shafarevich conjecture, which is perhaps a little more approachable. A positive-dimensional component \(D_\nu\) of the orbifold locus is called exceptional if \(\pi_1(D_\nu)\) has finite image in the orbifold fundamental group. The Shafarevich conjecture for Kähler orbifolds would imply that the image of \(\pi_1(\bigcup D_\nu)\) will be finite also, where \(\bigcup D_\nu\) is any connected union of exceptional components. In the opposite direction, cases of the Shafarevich conjecture can be deduced from statements of this type. Suppose that \(X\) is a Zariski open subset of a Kähler manifold, and that \(X\) has the property that whenever \(f: Z\to X\) is a holomorphic map from a compact complex space, then the image \(P\) of \(\pi_1(Z)\) in \(\pi_1(X)\) is finite if and only if every finite-dimensional representation of \(P\) has finite image. (Neither the author nor the reviewer is aware of any \(X\) for which this fails.) Then it is shown here that the universal covering manifold of \(X\) is holomorphically convex.