# Hodge Theory

Prerequisite: Complex analysis

Elliptic integrals. Abelian and multiple integrals. Basics of singular homology. Isomorphism of Leray-Thom-Gysin. Lefschetz’s theorem in hyperplane sections. Decomposition of Lefschetz. Lefschetz’s difficult theorem (enunciated). Ehresmann Fusion Theorem. Monodromy and evanescent cycles. Hodge’s conjecture and Lefschetz’s (1.1) theorem. Rham’s cohomology of soft hypersurfaces (Griffiths’s theorem). Hodge cycles of Fermat varieties. Picard number of Fermat surfaces. Hypercohomology. Differential forms and vector fields. Rham algebraic cohomology. Theorem of Atiyah-Hodge. Hodge filtration. Algebraic and analytical Gauss-Manin connection. Griffiths’ transversality theorem. Infinitesimal variation of Hodge structures (IVHS). Map of Kodaira-Spencer. Theorem of Noether-Lefschetz. Loter of Noether-Lefschetz and Hodge. 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