Dénombrement des types de K-homotopie. Théorie de la déformation
Effective finite generation for adjoint rings
We describe a bound on the degree of the generators for some adjoint rings on surfaces and threefolds.
Elliptic curves and canonical subgroups of formal groups.
Elliptische Funktionenkörper mit schlechter Reduktion.
Extending extremal contractions from an ample section.
Extremal contractions from 4-dimensional manifolds to 3-folds
Extremal rays on higher dimensional projective varieties.
Fano manifolds and quadric bundles.
Generalized Mukai conjecture for special Fano varieties
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
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
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,...
Global minimal models for endomorphisms of projective space
We prove the existence of global minimal models for endomorphisms of projective space defined over the field of fractions of a principal ideal domain.
Groups of components of Néron models of jacobians
Inverting Reid's exact plurigenera formula.
extension of adjoint line bundle sections
We prove an extension theorem of Ohsawa-Takegoshi type for line bundle sections on a subvariety of general codimension in a normal projective variety. Our method of proof gives conditions to be satisfied for such extension in a general setting, while such conditions are satisfied when the subvariety is given by an appropriate multiplier ideal sheaf.
Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville
Let be a compact hyperkähler manifold containing a complex torus as a Lagrangian subvariety. Beauville posed the question whether admits a Lagrangian fibration with fibre . We show that this is indeed the case if is not projective. If is projective we find an almost holomorphic Lagrangian fibration with fibre under additional assumptions on the pair , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian...
Limits of log canonical thresholds
Let denote the set of log canonical thresholds of pairs , with a nonsingular variety of dimension , and a nonempty closed subscheme of . Using non-standard methods, we show that every limit of a decreasing sequence in lies in , proving in this setting a conjecture of Kollár. We also show that is closed in ; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check...
Minimal models and the Kodaira dimension of algebraic fiber spaces.
Minimal models for elliptic curves with complex multiplication
Minimal models of algebraic threefolds : Mori's program