Geometry of the holomorphic symplectic manifolds and the characteristic foliation

In this thesis we study the characteristic foliation on a hypersurface in a smooth projective holomorphic symplectic manifold. Let us explain the problem in details. Let X be a smooth projective irreducible holomorphic symplectic manifold of dimension 2n and Y be a smooth hypersurface in X. Let σ be a holomorphic symplectic form on X. At every point x of Y the holomorphic symplectic form σ restricts to the tangent space ⊤ᵧ,ₓ of Y at x as an alternating form of corang 1. Thus, it has one-dimensional kernel. The characteristic foliation F of Y is the kernel of symplectic form σ restricted to Y. One can ask what could be the dimension of the Zariski closure of a general leaf of F. In this thesis we find the answer to this question in few cases.In the first case the irreducible holomorphic symplectic manifold X is equipped with a Lagrangian fibration π : X →Pn. One calls a hypersurface Y vertical if there exists a hypersurface D in ℙⁿ such that Y is the pre-image of D. We proved that the Zariski closure of a general leaf of the chatacteristic foliation on Y is dense in a fiber of the Lagrangian fibration π.In the second we consider a nef and big hypersurface Y in X. We prove that a general leaf of the characteristic foliation on Y is Zariski dense in Y. Afterwards, we study the characteristic foliation on singular hypersurfaces. We give few examples of singular vertical hypersurfaces such that a general leaf of the characteristic foliation is not Zariski dense in a fiber of the Lagrangian fibration. Next, we consider the variety X of lines in the cubic fourfold. We give two example of a singular hypersurface Y in X such that a general leaf of the characteristic foliation on Y is not Zariski dense in Y. Towards the end of the thesis we study the holomorphic symplectic fourfold constructed by O. Debarre and C. Voisin. We find a hypersurface in the manifold and construct a natural foliation of rank one on this hypersurface. We conjecture that this foliation is characteristic.