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Generalized Mukai conjecture for special Fano varieties

Marco Andreatta, Elena Chierici, Gianluca Occhetta (2004)

Open Mathematics

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.

Geography of log models: theory and applications

Vyacheslav Shokurov, Sung Choi (2011)

Open Mathematics

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.

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana, Thomas Peternell (2011)

Bulletin de la Société Mathématique de France

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 X have a pseudo-effective (instead of generically nef) determinant. A first consequence is that X 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 X is not covered by compact positive-dimensional analytic subsets,...

Global minimal models for endomorphisms of projective space

Clayton Petsche, Brian Stout (2014)

Journal de Théorie des Nombres de Bordeaux

We prove the existence of global minimal models for endomorphisms φ : ℙ N → ℙ N of projective space defined over the field of fractions of a principal ideal domain.

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