Hodge Theory, 17 August- 06 December 2020

Monday, Wednesday 9:00 (Rio de Janeiro's time)


The talks are available in the YouTube play list Videos
Some background will be taken from


  • 17/08/2020: Lecture 1
    Section 2.1 and Section 2.2 of [Mov]. Introduction to the history of elliptic/abelian/multiple integrals (periods). What is the Hodge conjecture (one of the Millennium Prize Problems)?
    I realized that right in the first day of my lectures, there will be no electricity (and so no internet at my apt.). Therefore, the first lecture will be from my ipad which I can use it in any place with internet!
  • 19/08/2020, Lecture 2 Chapter 2 of [MovVil]. Cech Cohomology relative to a covering, p-cochains skew symmetric p-cochains.
  • 24/08/2020, Lecture 3 Chapter 2 of [MovVil]. A fast review of the proof of \(\delta\circ \delta=0\). The 0-th cohomology=global sections. Section 2.6 Acyclic covering, Leray's Lemma (announcement).Acyclic covering for the constant sheaf \(\mathbb Z\). Good cover for manifolds (Bott-Tu page 42). Relative cohomology. Computing the cohomology of sphere: EXERCISE 1:
    $$ H^m(\mathbb S^n,\mathbb Z) \cong \left\{ \begin{array}{cc} \mathbb Z, & \, \, \hbox{ if } m=0,n\\ 0 & \, \, \hbox{ otherwise } \end{array} \right. , $$
    $$ H^m(\Bbb B^{n}, \Bbb S^{n-1},\mathbb Z) \cong \left\{ \begin{array}{cc} \mathbb Z, & \, \, \hbox{ if } m=n\\ 0, & \, \, \hbox{ otherwise } \end{array} \right. . $$
  • 26/08/2020, >Lecture 4 Section 5.2 of [Mov], EXERCISE 2: Exercise 5.1 of [Mov] Smooth projective varieties, two projective varieties intersects each other transversely, Hyperplane section, Veronese embedding and hypersurface sections, Theorem 5.1 the fundamental theorem of the topology of algebraic varieties. The case \(F/G^m\)
  • 31/08/2020, Lecture 5, Lefschetz hyperplane section theorem[section 5.4]. Theorems from topology needed to prove this: Gysin sequence (Section 4.6 [Mov]),
  • 02/09/2020, Lecture 6. The proof of the fundamental theorem of the topology of algebraic varieties I. Lefschetz fibration, Ehresmann's fibration
  • 07/09/2020. Lecture 7. Independence day of Brasil, The lecture will happen exceptionally 08/09/2020. The proof of the fundamental theorem of the topology of algebraic varieties II. Milnor number, Milnor fibration, Ehresmann's fibration theorem, Blow-up along the indeterminacy points of a Lefschetz pencil.
  • 09/09/2020, Lecture 8. Chapter 6 of [Mov]. Cohomology of smooth hypersurfaces, Picard-Lefschetz theory, Picard-Lefschetz formula.
  • 14/09/2020, Lecture 9. Zarsiki topology, affine varieties, differential forms (Section 10.3 [Mov])), Hypercohomology(Section 3.2 of [MovVil])
  • 16/09/2020, Lecture 10. Elements of algebraic de Rham cohomology, hypercohomology of the complex of C^infty forms is the usual de Rham cohomology, Serre's theorem, hypercohomology of the complex of algebraic forms for an affine variety is given by global forms, Atiyah-Hodge theorem (Hironaka's desingularization, Serre's GAGA). Obs. In the definition of normal crossing divisor in the video of Lecture 10 one must replace \(z_1=z_2=..=z_r=0\) with \(z_1=0 \cup z_2=0 \cup ...z_r=0 \)
  • 21/09/2020, Lecture 11. The idea of the proof of Atiyah-Hodge theorem, Section 4.3 of [Mov-Vil]. What is a quasi-isomorphism. De Rham cohomology of affine varieties given by tame polynomials (Brieskorn module H', Theorem 10.1) Chapter 10 of [Mov].
  • 23/09/2020, Lecture 12. Algebraic de Rham cohomology is isomorphic to the classical de Rham cohomology. Partition of unity. Hodge filtration.
  • 28/09/2020, Lecture 13. Hodge Filtration, Hodge decomposition, cup product.
  • 30/09/2020, Lecture 14. Polarization, top cohomology, trace map.
  • 05/10/2020, Lecture 15. Cohomology class of an algebraic cycle. Hodge cycle. Hodge conjecture.
  • 07/10/2020, Lecture 16. Chern class of line bundles and divisors, Lefschetz (1,1) theorem (Chapter 9 [Mov])
  • 19/10/2020, Lecture 17. Noether-Lefschetz theorem (Chapter 14 [Mov]).
  • 21/10/2020, Lecture 18. Residue map, Griffiths theorem on the de Rham cohomology of hypersurfaces, Restriction to an affine chart.
  • 26/10/2020, Lecture 19. Tame polynomial, discriminant, Brieskorn module.
  • 28/10/2020, Lecture 20. Gauss-Manin system.
  • 04/11/2020, Lecture 21. Hodge filtration and periods of the Fermat variety.
  • 09/11/2020, Lecture 22. Computing Hodge cycles of the Fermat variety.
  • 11/11/2020, Lecture 23. Periods of Hodge cycles of the Fermat variety and transcendence properties of the values of Gamma function.
  • 16/11/2020, Lecture 24. Hodge locus.
  • 18/11/2020, Lecture 25. Zariski tangent space of the Hodge locus passing through Fermat.
  • 23/11/2020, Lecture 26. General components of the Hodge locus, Infinite number of general components of the Noether-Lefschetz locus passing through Fermat, general and special components.
  • 25/11/2020, Lecture 27. Algebraic cycles, Villaflor's formula of periods of complete intersection algebraic cycles.