Algebraic approximation of Kähler threefolds (Q2909661)

Given a compact complex manifold, an algebraic approximation of \(X\) is a proper holomorphic submersion \(\pi: \mathcal{X}\rightarrow T\) such that \(\mathcal{X}_0:=\pi^{-1}(0)\cong X\) and such that there exists a sequence \((t_k)_{k\in \mathbb{N}}\subset T\) converging to \(0\) with \(\mathcal{X}_{t_k}\) projective for all \(k\). It is known that every compact Kähler surface is algebraically approximable and that this is false in dimension \(\geq 4\) [\textit{C. Voisin}, Invent. Math. 157, No. 2, 329--343 (2004; Zbl 1065.32010)].NEWLINENEWLINEThis paper starts to address the problem for compact Kähler threefolds. More in details the author shows that the following classes of such threefolds are algebraicaly approximable: projective bundles \(\mathbb{P}(V)\) associated to holomorphic rank-two vector bundles \(V\) over compact Kähler surfaces with Kodaira dimension \(0\), threefolds bimeromorphic to \(\mathbb{P}^1\times S\) for a compact Kähler surface \(S\), conic bundles \(f:X\rightarrow S\) over a K3 surface such that \(f_*(K^*_{X/S})\) splits as a direct sum of line bundles and conic bundles \(f:X\rightarrow S\) over a compact Kähler surface with an elliptic fibration \(r:S\rightarrow C\) over a smooth curve such that \(f_*(K^*_{X/S})\) is trivial when restricted to the general fiber of \(r\).NEWLINENEWLINEAn intermediate step in the proof is the study of the existence of infinitesimal deformations for every conic bundle with relative Picard number \(1\) over a non-algebraic Kähler compact surface \(S\). It is proven that there exist positive-dimensional families in all cases apart from two exceptions.