On non-Kählerianity of nonuniform lattices in \(\mathrm{SO}(n,1)\) \((n\geq 4)\)
The question is posed whether an abstract group can be fundamental group of a compact Kähler or quasi-compact Kähler (obtained by deleting a normal crossing divisor) manifold? The answer is ``no'' for any cocompact lattice in \(\mathrm{SO}(n,1)\), \(n\geq 3\), [\textit{J. A. Carlson} and \textit{D. Toledo}, Publ. Math., Inst. Hautes Étud. Sci. 69, 173--201 (1989; Zbl 0695.58010)]. Another negative result is obtained in this paper, namely: Let \(\Gamma\) be a nonuniform lattice of \(\mathrm{SO}(n,1)\), \(n\geq 4\). Let \(M\) be any compact Kähler manifold and let \(D\) be any normal crossing divisor of \(M\). Then \(\Gamma\), as an abstract group, is not isomorphic to \(\pi _1(M \setminus D)\). The theorem is true in the case if \(M \setminus D)\) is quasi-projective variety according to the well known Hironaka's theorem. The proposed proof uses the infinite energy harmonic maps theory of \textit{J. Jost} and \textit{K. Zuo} [J. Differ. Geom. 47, No. 3, 469--503 (1997; Zbl 0911.58012)].
scientific article; zbMATH DE number 1556662
scientific article