Prerequisite: Complex analysis

Elliptic integrals. Abelian and multiple integrals. Basic notions of singular homology. Leray-Thom-Gysin isomorphism. Lefschetz theorem on hyperplane sections. Lefschetz decomposition. Difficult Lefschetz theorem (statement). Ehresmann’s fibration theorem. Monodromy and evanescent cycles. Hodge’s conjecture and Lefschetz’s (1,1) theorem. Rham cohomology of smooth hypersurfaces (Griffiths theorem). Hodge cycles of Fermat varieties. Picard number of Fermat surfaces. Hypercohomology. Differential forms and vector fields. Algebraic Rham cohomology. Atiyah-Hodge theorem. Hodge filtration. Algebraic and analytic Gauss-Manin connection. Griffiths transversality theorem. Infinitesimal variation of Hodge structures (IVHS). Kodaira-Spencer map. Noether-Lefschetz theorem. Noether-Lefschetz and Hodge loci. Tangent spaces of Hodge loci.

References:
LEWIS, JAMES D,. – A survey of the Hodge conjecture. Appendix B by B. Brent Gordon. CRM Monograph Series, 10. American Mathematical Society, Providence, RI, 1999.
CLAIRE VOISIN. – Hodge theory and complex algebraic geometry. Volume 76 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002.
CLAIRE VOISIN. – Hodge theory and complex algebraic geometry. {II}Volume 77 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2003.
HOSSEIN MOVASATI. A course in Hodge theory: with emphasis on multiple integrals, Lecture notes, http://w3.impa.br/%7Ehossein/myarticles/hodgetheory.pdf

 

* Basic syllabus. The teacher has the autonomy to make any changes.