Ph.D. level mini-course at IMPA.
Contents: 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.
Reference: Periods of Algebraic Cycles, by R. Villaflor. IMPA Ph.D. Thesis.
Period and time: September 11 to 27, 2018. Tuesday 15h30-17h00 (room 333), Wednesday 14h00-15h00 (room 333), Thursday 10h30-12h00 (room 347).
Lecture 1: Algebraic de Rham cohomology and Hodge filtration for projective varieties.
Lecture 2: Atiyah-Hodge theorem and Hodge filtration for affine varieties.
Lecture 3: Residues of meromorphic forms and Griffiths basis theorem.
Lecture 4: Volume forms and pull-back in algebraic de Rham cohomology.
Lecture 5: Periods of linear algebraic cycles.
Lecture 6: Periods of complete intersection algebraic cycles.
Lecture 7: The Gauss-Manin connection.
Lecture 8: Infinitesimal variations of Hodge structures and Hodge locus.
Lecture 9: Variational Hodge conjecture.