Hodge Theory, 17 August 06 December 2020
Monday, Wednesday 9:00 (Rio de Janeiro's time)
meet.google.com/vmroomeurw
The talks are available in the YouTube play list
Videos
References:
 [VoisinI] and [VoisinII] C. Voisin, Hodge Theory and Complex Algebraic Geometry, Cambridge University Press, I, II, 2003
 [Mov] H. Movasati, A Course in Hodge Theory:
With Emphasis on Multiple Integrals, Soon will be published by International Press, Boston

[MovVil]
H. Movasati, R. Vilaflor,
A Course in Hodge Theory: Periods of Algebraic cycles
Some background will be taken from
 W.S. Massey, A basic course in algebraic topology, 1991, GTM, Springer.
 G. E. Bredon, Topology and Geometry, GTM, 1993
 R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, GTM, 1982

Prerequisite: Complex analysis and basic topology. A basis knowledge of Algebraic Geometry and Algebraic Topology would be useful,
however, I believe that such a basic knowledge can be learned during this course!

Final grades: 50% Exercises (there will be around 50 exercises) 50% Expositions. Those who need grade are supposed
to prepare two or three half an hour exposition of parts of the course starting from 28 September 2020.

Language: I will start the lectures in English, but the audience is allowed to ask questions in other languages (Portuguese, Spanish, Persian etc).
I will repeat the question in English for the whole audience. I might also switch to Portuguese for few minutes after a question.

Each lecture will be around 60 minutes (recorded). Apart from this there will be 30 minutes of discussion of exercises,
the content of previous lectures, etc (not recorded).

I will use my webcam and write in front of it. For a sample, see my seminar in
GADEPs.
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, pcochains skew symmetric pcochains.
24/08/2020,
Lecture 3
Chapter 2 of [MovVil]. A fast review of the proof of \(\delta\circ \delta=0\). The 0th cohomology=global sections. Section 2.6 Acyclic covering,
Leray's Lemma (announcement).Acyclic covering for the constant sheaf \(\mathbb Z\). Good cover for manifolds (BottTu page 42).
Relative cohomology.
Computing the cohomology of sphere: EXERCISE 1:
$$
H^m(\mathbb S^n,\mathbb Z) \cong
\left\{
\begin{array}{cc}
\mathbb Z, & \, \, \hbox{ if } m=0,n\\
0 & \, \, \hbox{ otherwise }
\end{array}
\right. ,
$$
$$
H^m(\Bbb B^{n}, \Bbb S^{n1},\mathbb Z) \cong
\left\{
\begin{array}{cc}
\mathbb Z, & \, \, \hbox{ if } m=n\\
0, & \, \, \hbox{ otherwise }
\end{array}
\right. .
$$
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, Blowup along the indeterminacy points of a Lefschetz pencil.
09/09/2020,
Lecture 8.
Chapter 6 of [Mov].
Cohomology of smooth hypersurfaces, PicardLefschetz theory, PicardLefschetz 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, AtiyahHodge 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 AtiyahHodge theorem, Section 4.3 of [MovVil]. What is a quasiisomorphism.
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. NoetherLefschetz 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. GaussManin 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 NoetherLefschetz 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.