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Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.
This is an introduction to geography of log models with applications to positive cones of Fano type (FT) varieties and to geometry of minimal models and Mori fibrations.
We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold have a pseudo-effective (instead of generically nef) determinant. A first consequence is that is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of is not covered by compact positive-dimensional analytic subsets,...
We prove the existence of global minimal models for endomorphisms of projective space defined over the field of fractions of a principal ideal domain.
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