Let X be an irreducible holomorphic symplectic fourfold and D a smooth hypersurface in X. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction of the symplectic form to D) is not algebraic unless D is uniruled. Suppose now that the Zariski closure of its general leaf is a surface. We prove that X has a lagrangian fibration and D is the inverse image of a curve on its base.

В работе представлена верхняя оценка числа решений алгебраического уравнения P(x,y)=0 над полем F_p, принадлежащих подгруппе G мультипликативной группы F_p^*.

We apply the technique of the paper "The abelian/nonabelian correspondence and Frobenius manifolds" by I. Ciocan-Fontanine, B. Kim, C. Sabbah to construct Saito primitive forms for Gepner singularities.