On the Kodaira problem for uniruled Kähler spaces (Q2210771)

Let \(X\) be a compact complex variety and let \(\pi: \mathscr{X}\to S\ni 0\) be a deformation of \(X\) with \(\mathscr{X}_0\simeq X\). \(\pi\) is called an \textit{algebraic approximation} of \(X\) if there is a sequence \((s_n)_{n\in\mathbb{N}}\) in \(S\) tending to \(0\) such that each \(\mathscr{X}_{s_n}\) is a projective variety. In the 1960s, Kodaira proved that every compact Kähler surface admit an algebraic approximation [\textit{K. Kodaira}, Ann. Math. (2) 78, 1--40 (1963; Zbl 0171.19601)], and based on this observation he asks if every compact Kähler manifold admits an algebraic approximation, known as the \textit{Kodaira problem}. In her 2004 (resp. 2006) works [Invent. Math. 157, No. 2, 329--343 (2004; Zbl 1065.32010)], resp. [J. Differ. Geom. 72, No. 1, 43--71 (2006; Zbl 1102.32008)], \textit{C. Voisin} gave a negative answer to this problem (resp. bimeromorphic version of this problem) by constructing a Kähler manifold that is not (whose smooth bimeromorphic models are not) of the homotopy type of any projective manifold. However Voisin's works do not close the study of the bimeromorphic Kodaira problem in the sense that from the viewpoint of the minimal model program (MMP for short) it is natural to include the singular models of a compact Kähler manifold into consideration, and to ask whether the output of an MMP of a compact Kähler manifold admits an algebraic approximation. Hence one may consider the following two problems according to the dichotomy: \begin{itemize} \item For a non-uniruled Kähler manifold \(X\), the output of an \(K_X\)-MMP (if exists, as expected) is a minimal Kähler varieties \(X_{\text{min}}\), then one may ask whether it admits an algebraic approximation. This is indeed a conjecture of Th. Peternell. \item For a uniruled Kähler manifold \(X\), the output of a \(K_X\)-MMP (if exists, as expected) is a Mori fibre spaces (MFS, for short) \(X\to Y\). Similarly we can ask the question about the existence of an algebraic approximation for \(X'\). In this case, the problem is a bit more complicated: the algebraic approximability of \(X'\) relates to that of \(Y\). Then a natural question is: for a MFS \(X'\to Y\), does the algebraic approximability of \(Y\) implies that of \(X'\)? Though it has been confirmed by the work of H.-Y. Lin that uniruled Kähler threefolds are algebraically approximable, the question is not true in general as shown by the article under review. \end{itemize} More precisely, the authors of the article construct the following example which arises from Voisin's 2006 work. First recall the construction of Voisin. For \(n\ge 4\), let \(T\) be a \(n\)-dimensional scenic torus, that is, a \(n\)-dimensional complex torus endowed with an endolorphism \(\phi\) such that the eigenvalues of the induced morphism \(\phi_\ast: \text{H}_1(T,\mathbb{C})\to\text{H}_1(T,\mathbb{C})\) are pairwise distinct and non-real, and the Galois group of the field generzted by them over \(\mathbb{Q}\) is the full symmetric group. Such tori exist as shown in [\textit{B. L. Van der Waerden}, Algebra I. Unter Benutzung von Vorlesungen von E. Artin und E. Noether. Mit einem Geleitwort von Jürgen Neukirch. Berlin: Springer-Verlag (1993; Zbl 0781.12002)] (see Theorem 3.2). Let \(T\check{}\) be the dual torus of \(T\), and let \(\mathscr{P}\) be the normalized Poincaré line bundle on \(T\times T\check{}\). Set \(\mathscr{P}_\phi:=(\phi,\text{id}_{T\check{}})^\ast\mathscr{P}\) and \(\mathscr{E}_\phi:=\mathscr{P}_\phi\oplus\mathscr{P}_\phi^{-1}\). And consider the automorphisms \(i_\phi\) and \(\hat i_\phi\) on \(\mathbb{P}(\mathscr{E}_\phi)\) induced by \((-\text{id}_T,\text{id}_{T\check{}})\in\text{Aut}(T\times T\check{})\) and \((\text{id}_T,-\text{id}_{T\check{}})\in\text{Aut}(T\times T\check{})\) respectively. Set \(\mathscr{Q}_\phi:=\mathbb{P}(\mathscr{E}_\phi)/\langle i_\phi,\hat i_\phi\rangle\). Note that \(\mathscr{P}_{\text{id}}=\mathscr{P}\) and we drop the subscript in the above notation for \(\phi=\text{id}\). Set \(Z:=\mathbb{P}(\mathscr{E})\times_{T\times T\check{}}\mathbb{P}(\mathscr{E}_\phi)\) and let \(G\) be the group generated by the automorphisms \((i,i_\phi)\) and \((\hat i, \hat i_\phi)\). Set \(X:Z/G\). By Voisin's 2006 paper, any smooth bimeromorphic model of \(X\) does not have the homotopy type of a complex projective manifold, thus is not algebraically approximable. By a local calculation, it is shown that \(X\) is a \((2n+2)\)-dimensional Kähler space with only terminal quotient singularities of codimension \(n+1\). In particular, by [\textit{J. Stevens}, Deformations of singularities. Berlin: Springer (2003; Zbl 1034.14003)] \(X\) has rigid singularities (Definition 2.5). Then by taking a functional resolution as constructed in [\textit{J. Kollár}, Lectures on resolution of singularities. Princeton, NJ: Princeton University Press (2007; Zbl 1113.14013)] (see Theorem 2.1), one sees immediately that \(X\) is not algebraically approximable. Moreover, by some analytic version of the rigidity lemma, one can show that \(X\) is `minimal' in the sense that any bimeromorphic map from \(X\) to a normal complex variety is an isomorphism and that every run of the \(K_X\)-MMP yields immediately a Mori fibre space which is either \(X\to\mathscr{Q}\) or \(X\to\mathscr{Q}_\phi\). However, from the algebraic approximability of complex tori one can easily deduce that \(\mathscr{Q}\) is algebraically approximable.