Hodge Theory, 17 August- 06 December 2020

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

meet.google.com/vmr-oome-urw

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





Lectures

  • 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, de Rham cohomology of affine varieties given by tame polynomials (Brieskorn modules)