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Calabi-Yau manifolds with large Picard number. (English) Zbl 0688.14032
A Calabi-Yau manifold is a smooth complex projective threefold V with trivial canonical bundle and no global 1-forms or 2-forms. This paper is dealing with C-Y manifolds with Picard number \(\rho (V)>19.\)
The author defines a Calabi-Yau model as a positive 3-fold \(\bar V\) with only canonical singularities of \(index\quad 1\) for which there is a resolution of singularities \(\pi:\quad V\to \bar V\) with V a C-Y manifold and he proves that any C-Y manifold is a resolution of a C-Y model \(\bar V\) with Picard number \(\rho(\bar V)\leq 19\). - The main tools for the proof are:
(i) Mori’s techniques for the classification of 3-folds, in particular the analysis of the singularities arising from small contractions;
(ii) the study of the cubic hypersurface \(W\subset {\mathbb{P}}^{\rho -1}\) consisting of points representing \(divisors\quad D\) with \(D^ 3=0\). In fact the existence of the required contraction is reduced to the existence of a rational point on W, satisfying some suitable conditions.
Reviewer: L.Picco Botta

14J30 \(3\)-folds
14E30 Minimal model program (Mori theory, extremal rays)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
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