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Smoothing 3-folds with trivial canonical bundle and ordinary double points. (English) Zbl 0829.32012

Yau, Shing-Tung (ed.), Essays on mirror manifolds. Cambridge, MA: International Press. 458-479 (1992).
Let \(X\) be a smooth threefold with trivial canonical bundle and disjoint rational curves \(C_1, \dots, C_l\) of type \((-1, 1)\), then one can contact those rational curves to obtain a singular threefold \(X_0\) with \(l\) ordinary double points. Here, by type \((-1, 1)\), we mean that the normal bundle of each \(C_1, \dots, C_l\) is a direct sum of two tautological line bundles on the rational curve. A natural question is whether or not there exists a smoothing of \(X_0\), in other words, there exists a complex fourfold \(\mathcal X\), together with a proper flat map \(\pi : {\mathcal X} \mapsto \Delta\), where \(\Delta\) is the unit disk in \(C\), such that \(\pi^{-1} (0) = X_0\) and \(\pi^{-1} (t) = X_t\) is smooth for \(t\neq 0\). It is shown by R. Friedman that there is an infinitesimal smoothing of \(X_0\) if and only if the fundamental classes \([C_i]\) in \(H^2 (X; \Omega^2_X)\) satisfy a relation \(\sum_i \lambda_i [C_i] = 0\) such that, for every \(i\), \(\lambda_i \neq 0\). This infinitesimal smoothing can be realized by a real smoothing in case the obstruction group \({\mathbf T}^2_{X_0}\) happens to be zero.
The purpose of this paper is to show that the infinitesimal smoothing can always be ralized by a real smoothing. The main result is the following.
Theorem 0.1. Let \(X_0\) be a singular threefold with \(l\) ordinary double points as only singular points \(p_1, \dots, p_l\). Let \(X\) be the small resolution of \(X_0\) by replacing \(p_i\) by a smooth rational curve \(C_i\). Assume that \(X\) is Kähler and has trivial canonical line bundle. Furthermore, we assume that the fundamental classes \([C_i]\) in \(H^2(X, \Omega^2_X)\) satisfy a relation \(\sum_i \lambda_i [C_i] = 0\) such that, for every \(i\), \(\lambda_i \neq 0\). Then \(X_0\) admits a smoothing.
For the entire collection see [Zbl 0816.00010].

MSC:

32J17 Compact complex \(3\)-folds
14J30 \(3\)-folds
32S45 Modifications; resolution of singularities (complex-analytic aspects)
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