An introduction to algebraic de Rham cohomology and infinitesimal variations of Hodge structures

The mini course will be divided into three parts. In the first part, we will give an introduction to algebraic de Rham cohomology and Hodge filtration for projective and affine varieties.
We will prove Atiyah-Hodge’s theorem and Griffiths’ theorem on the basis for the middle cohomology of hypersurfaces. The second part will be devoted to computations of periods of algebraic cycles inside hypersurfaces. In the third part, we will introduce Infinitesimal Variations of Hodge Structures using the algebraic description of Gauss-Manin connection given by Katz-Oda. Finally, we will introduce Hodge and Noether-Lefschetz loci and we will show how the periods of algebraic cycles can be used to study the local components of these spaces and prove some cases of the Variational Hodge conjecture.