Diagram Method for 3-Folds and its Application to Kahler Cone and Picard Number of Calabi-Yau 3-Folds, I. with Appendix by Vyacheslav V. Shokurov: "Anticanonical boundedness for curves"

Abstract

We prove the general diagram method theorem valid for the quite large class of 3-folds with Q-factorial singularities (see Basic Theorem 1.3.2 and also Theorem 2.2.6). This gives the generalization of our results about Fano 3-folds with Q-factorial terminal singularities (Preprint alg-geom/9311007). As an application, we get the following result about 3-dimensional Calabi-Yau manifolds X: Assume that the Picard number \rho (X) > 93. Then one of two cases (i) or (ii) holds: (i) there exists a small extremal ray on X; (ii) There exists a nef element h such that h^3=0 (thus, the nef cone NEF(X) and the cubic intersection hypersurface W_X have a common point; here, we don't claim that h is rational!). As a corollary, we get: Let X be a 3-dimensional Calabi-Yau manifold. Assume that the nef cone NEF(X) is finite polyhedral and X does not have a small extremal ray. Then there exists a rational nef element h with h^3=0 if \rho(X)>93. To prove these results about Calabi-Yau manifolds, we also use one result of V.V. Shokurov on the length of divisorial extremal rays (see Appendix by V.V. Shokurov). Thus, one should consider the results on Calabi-Yau 3-folds above as our common results with V.V. Shokurov. We also discuss generalization of results above to so called Q-factorial models of Calabi-Yau 3-folds, which sometimes permits to involve non-polyhedral case and small extremal rays to the game.