On manifolds with trivial logarithmic tangent bundle
The automorphism group \(\text{Aut}(X)\) of a compact complex manifold \(X\) is a finite dimensional complex Lie group with Lie algebra the holomorphic vector fields on \(X\). Since the vector fields are tangent to the Aut\((X)\)-orbits, a connected compact complex manifold \(X\) with trivial holomorphic tangent bundle \(T(X)\) is a quotient of \(\text{Aut}(X)\) by a uniform discrete subgroup. If \(X\) is Kähler, then \(X=\text{Aut}^0(X)\) is a compact complex torus [see \textit{H. C. Wang}, Proc. Am. Math. Soc. 5, 771--776 (1954; Zbl 0056.15403)]. The author considers the situation of a non-compact complex manifold \(U\) with holomorphically trivial tangent bundle \(T(U)\) which is Zariski-open in a compact complex manifold \(X\) of class \({\mathcal C}\), i.e. \(X\) is bimeromorphic to a compact Kähler manifold. If \(D:=X\backslash U\) is a divisor with only simple normal crossings the analytic sheaf \(\Omega^1(\log D)\) of logarithmic 1-forms with respect to \(D\) on \(X\) is locally free. Its dual bundle, the logarithmic tangent bundle \(T(-\log D)\), is an extension of \(T(U)\) to \(X\) [cf. \textit{D. Mumford}, Invent. Math. 42, 239--272 (1977; Zbl 0365.14012)]. The main result of the paper under review is the equivalence of the conditions (1) ``\(D\) is a divisor with only simple normal crossings and \(T(-\log D)\) is holomorphically trivial'', and (2) ``\(\text{Aut}(X)\) contains a complex semi-torus \(G\) such that \(G\) operates freely and transitively on \(U\) and all isotropy subgroups of \(G\) in \(D\) are complex semi-tori as well''. Here a complex semi-torus is defined to be a quotient of \((\mathbb C^n,+)\) by a discrete subgroup \(\Gamma\) with \(\mathbb C\Gamma=\mathbb C^n\). The class \({\mathcal C}\) condition is needed in the proof \((1)\Rightarrow (2)\) to guarantee that the logarithmic 1-forms on \(X\) are closed. There are many examples of toric varieties with an equivariant compactification by a divisor \(D\) such that (1) is true, especially those which are in class \({\mathcal C}\) but not Kähler [see \textit{J. Varouchas}, Math. Ann. 283, 13--52 (1989; Zbl 0738.32005)]. The author shows by examples that in statement (2) the condition for the isotropy groups is necessary and that the situation can be totally different if \(X\) is not in class \({\mathcal C}\).
scientific article; zbMATH DE number 2111467