A compactification of \((\mathbb{C}^*)^4\) with no non-constant meromorphic functions (Q5956598)

Here the authors prove the following result which shows that smooth compactifications of \((\mathbb{C}^*)^n\) may be non-Moishezon, in sharp contrast with the conjectural behaviour of smooth compactifications of \(\mathbb{C}^n\).NEWLINENEWLINENEWLINETheorem 1. Let \(T\) be any two-dimensional complex torus. Then there exists a smooth compact Kähler 4-fold \(X(T)\) and a smooth divisor \(D\subset X(T)\) such that \(D\) is biholomorphic to \(\mathbb{P}^1\times T\) and \(X(T)\setminus D\) is biholomorphic to \((\mathbb{C}^*)^4\). For a general torus \(T\) the manifold \(X(T)\) has no non-constant meromorphic function.