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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.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J30 $$3$$-folds 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 14B05 Singularities in algebraic geometry
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