Global smoothing of Calabi-Yau threefolds. (English) Zbl 0861.14036

Let \(Z\) be a Calabi-Yau threefold, i.e. a projective threefold with only rational singularities and with \(K_Z \sim 0\). Assume that \(Z\) admits only isolated rational hypersurface singularities. Then \(Z\) can be deformed to a Calabi-Yau threefold \(Z'\) with only ordinary double points; \(Z'\) is smooth if \(Z\) is \(\mathbb{Q}\)-factorial. Furthermore if \(X\) is a normal projective threefold with only isolated rational hypersurface singularities and \(h^2(X, {\mathcal O}_X) =0\), then the rank of the abelian group of Weil divisors of \(X\) modulo the Cartier ones is expressed in terms of the Betti numbers for the singular cohomology of \(Z\). The smoothing problems here considered are faced by two different ways: one using the vanishing theorems by Guillèn, Navarro Aznar and Puerto; the other using an invariant introduced by the first author for an isolated rational singularity.


14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J30 \(3\)-folds
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14B05 Singularities in algebraic geometry
Full Text: DOI EuDML


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