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\(\Gamma\)-reduction for smooth orbifolds. (English) Zbl 1163.14020

The \(\Gamma\)-reduction of a compact Kähler manifold, also known as Remmert reduction or the Shafarevich map, is a meromorphic fibration of the universal cover such that (among other technical requirements) any compact analytic subset passing through a very general point has to be contained in a fibre. In this version it was constructed by Campana. Here we see an orbifold version, valid for smooth orbifolds \((X/\Delta)\): that is, \(X\) is a compact Kähler manifold and \(\Delta\) is a \({\mathbb Q}\)-divisor with normal crossings of the form \(\Delta =\sum (1-{{1}\over{m_j}})\Delta_j\), where \(m_j>1\) are integers. There is a notion of orbifold universal cover \(\tilde X_\Delta\), and the result is that there is a \(\Gamma\)-reduction with the same properties as in the manifold case.
The methods are those of Campana. What one needs is a suitable Kähler metric, and the difference here is that one must allow singular metrics with suitable singularities along \(\Delta\). The main point, then, is to give a local description of this metric \(\omega_\Delta\) and check that it has the right properties: one must first modify \(\Delta\) slightly if the branching of the universal covering map is not exactly \(\Delta\). Then if \(\Delta_j=(s_j=0)\), where \(s_j\in H^0({\mathcal O}_X(\Delta_j))\) and \(\omega\) is a Kähler metric on \(X\), then for any choice of smooth metric on \({\mathcal O}_X(\Delta_j)\) and for \(C\in{\mathbb R}\) positive enough we may take \[ \omega_\Delta =C\omega+\sum_j\sqrt{-1}\partial\bar\partial |s_j|^{2/m_j}. \]

MSC:

14J10 Families, moduli, classification: algebraic theory
14E20 Coverings in algebraic geometry
32J27 Compact Kähler manifolds: generalizations, classification
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