We will present some classification results for (smooth, complex) Fano 4-folds X with Picard number rho(X)>6. First of all, if rho(X)>9, then X is a product of del Pezzo surfaces; this is sharp, since we know one family of Fano 4-folds with rho(X)=9 that is not a product of surfaces. In the range rho(X)=7,8,9, we will explain some partial classification results, based on a detailed and...
I will report on a joint work-in-progress with F. Bastianelli. It is well known thanks to Ein that very general complete intersections of multidegree (d_1,…,d_c) in the projective n-space do not contain rational curves as soon as d_1+…+d_c > 2n-c-1. This result has been sharpened in the case of hypersurfaces thanks to a method introduced by Voisin that inspired further work by Clemens, Ran and...
The cone conjecture predicts the shape of the nef and movable cones of certain projective varieties, modulo automorphisms. It is notably known for K3 surfaces, abelian varieties, hyperkähler manifolds, and in a few sporadic cases. Together with standard MMP conjectures, it implies the finiteness of minimal models, which can be used to improve birational boundedness to boundedness for certain...
I will report on some recent progress on the construction of a moduli space of surface foliations of general type. Time permitting I will also discuss some aspects of this moduli problem in general, as well as appliclations to the construction of moduli of fibred varieties. This talk will partly cover joint work with S. Velazquez and R. Svaldi.
I shall discuss recent ongoing work on the boundedness of algebraically integrable foliations of general type based on the theory of adjoint foliated structures. In particular, I will discuss a birational boundedness theorem that builds on the proof of McKernan’s ACC conjecture for interpolated log canonical thresholds for algebraically integrable foliations.
A family of Calabi–Yau varieties (or, more generally, log Calabi–Yau pairs) f:X->Y naturally induces a moduli divisor M, measuring the variation of the family f. Based on earlier works of the Japanese school, Prokhorov and Shokurov conjectured that M is semiample. In this talk, we discuss a proof of this conjecture and, time permitting, some immediate applications in birational geometry. In...
My talk is based on my joint work with Professor Koji Fujiwara (Kyoto University) and Professor Xun Yu (Tianjing University).
Main result of my talk is the finiteness of the N\'eron-Severi lattices of complex projective K3 surfaces whose automorphism groups are non-elementary hyperbolic under the assumption that the Picard number greater than or equal to 6 which is optimal to ensure the...
Reeb’s local and global theorems are classical results in the theory of smooth foliations, giving conditions under which a foliation is locally or globally the pullback of a foliation by points. These results depend on assumptions such as compactness and the finiteness of the fundamental group or holonomy of a leaf. In contrast, the Kupka Theorem concerns the local study of singular foliations...
The notion of an Enriques manifold was introduced by Oguiso-Schröer as a complex manifold which is not simply connected and whose universal covering is an irreducible symplectic manifold. From the viewpoint of birational geometry, we want to understand its behaviour under Minimal Model Program (MMP) operations. Based on the result of Lehn-Pacienza showing that any MMP starting from a primitive...
In this talk, we discuss the boundedness problem for log Calabi–Yau fibrations whose bases and general fibers are bounded. We show that, after fixing certain natural invariants, the total spaces of such fibrations are bounded in codimension one. Furthermore, we prove that the total spaces themselves are bounded when the general fibers have vanishing irregularity. As an application, we obtain...
