En nous appuyant sur la conjecture de Bloch–Kato en K-théorie de Milnor, nous établissons un lien général entre le défaut de la conjecture de Hodge entière pour la cohomologie de degré $4$ et le troisième groupe de cohomologie non ramifiée à coefficients $\mathbb{Q}/\mathbb{Z}$. Ceci permet de montrer que sur un solide (en anglais, űthreefold») uniréglé le troisième groupe de cohomologie non ramifiée à coefficients $\mathbb{Q}/\mathbb{Z}$ s’annule, ce que la K-théorie algébrique ne permet d’obtenir que dans certains cas. Ceci permet à l’inverse de déduire d’exemples ayant leur source en K-théorie que la conjecture de Hodge entière pour la cohomologie de degré $4$ peut être en défaut pour les variétés rationnellement connexes. Pour certaines familles à un paramètre de surfaces, on établit un lien entre la conjecture de Hodge entière et l’indice de la fibre générique.

Building upon the Bloch–Kato conjecture in Milnor K-theory, we relate the third unramified cohomology group with $\mathbb{Q}/\mathbb{Z}$ coefficients with a group which measures the failure of the integral Hodge conjecture in degree $4$. As a first consequence, a geometric theorem of Voisin implies that the third unramified cohomology group with $\mathbb{Q}/\mathbb{Z}$ coefficients vanishes on all uniruled threefolds. As a second consequence, a 1989 example by Colliot-Thélène and Ojanguren implies that the integral Hodge conjecture in degree $4$ fails for unirational varieties of dimension at least $6$. For certain classes of threefolds fibered over a curve, we establish a relation between the integral Hodge conjecture and the computation of the index of the generic fiber.

We study the symplectic geometry of rationally connected $3$-folds. The first result shows that rational connectedness is a symplectic deformation invariant in dimension $3$. If a rationally connected $3$-fold $X$ is Fano or has Picard number $2$, we prove that there is a nonzero Gromov–Witten invariant with two insertions being the class of a point. That is, $X$ is symplectic rationally connected. Finally we prove that many rationally connected $3$-folds are birational to a symplectic rationally connected variety.

In this work we prove that the shallow water flow, subject to strong wind forcing and linearized around an adequate stationary profile, develops for large time closed trajectories due to the propagation of Rossby waves, while Poincaré waves are shown to disperse. The methods used in this paper involve semiclassical analysis and dynamical systems for the study of Rossby waves, while some refined spectral analysis is required for the study of Poincaré waves, due to the large time scale involved which is of diffractive type.

The goal of this paper is to construct infinite-dimensional Lie algebras by using infinite product identities and to use these Lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and Lie algebras. We expect the Lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory (yet to be fully constructed) with symmetries given by the monster simple group.

We introduce vector-valued modular functions attached to families of modular functions of different levels, and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half-planes. We recast this result using the Borcherds–Harvey–Moore singular theta-lift and show that the vector-valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps.

For each element of the monster simple group, we construct an infinite-dimensional Lie algebra, such that its denominator formula is an infinite product expansion of the automorphic function arising from that element’s McKay–Thompson series. These Lie algebras have the unusual property that their simple roots and all root multiplicities are known. We show that under certain hypotheses, characters of groups acting on these Lie algebras form functions on the upper half-plane that are either constant or invariant under a genus zero congruence group.