# Hodge theory, Fall 2010

Organizational: The talks will be 2x45 minutes with a short break. Time and place: Fridays 10:30 AM in Room 312.

The basic assumption throughout this semester is that we are going to work exclusively with varieties over the complex numbers. In the first couple of lectures, I am going to give a brief introduction to Hodge structures and why the cohomology groups H^i(X, Z) of a smooth projective algebraic variety X over the complex numbers carries such a hodge structure. The idea is that in the following lectures we will simply use the existence and functoriality to see what one can do with this. I will also briefly talk about the Hodge conjecture and Lefschetz (1,1), etc.

On this page we will post the schedule of lectures. For now I am going to post the topics here. I want you (= the graduate students) pick a topic you would like to lecture about and email me your preferences. I'll make it an ordered list, but this is not necessarily the order in which the talks will be given. Also, some of the talks may be split into two.

Note, note, note! I still have to find more literature for each topic. Moreover, a great resource right here in the department is Bob Friedman, who is much more familiar with this material then I am!

1. Introduction to Hodge structures and Hodge structure on cohomology
Speaker: Johan de Jong.
Date: Friday, September 10.
2. Elementary observations about Hodge structures of algebraic varieties
Speaker: Johan de Jong.
Date: Friday, September 17.
3. Curves, H^1 of a curve, the Jacobian
Speaker: Alon Levy.
Date: Friday, September 24.
This might be a good first lecture to get started. Explain how the H^1 of a curve determines a principally polarized abelian variety and show that this abelian variety can serve both as the Albanese variety of the curve, as well as the Picard variety.
4. Torelli theorem for curves
Speaker: Jie Xia.
Date: Friday, October 1.
Of course this is a bit of a cheat, because Torelli is a very geometric thing in this case, and you do not need to use Hodge theory (and you can extend the result to positive characteristic). Still it is, in several senses, the most basic example of Torelli as the name suggests. Possible reference:
• Milne, J. S. Jacobian varieties Arithmetic geometry (Storrs, Conn., 1984), 167--212, Springer, New York, 1986.
5. Lecture on Abel-Jacobi maps.
Speaker: Hang Xue.
Date: Friday, October 8.
Define intermediate Jacobians. Define the Abel-Jacobi map, say for curves on a threefold. Compute an example, for example take the Fermat quintic threefold. There exists a cone of lines on this. Show that the image of the abel Jacobi map gives a proper sub-hodge structure of H^3. Ask Bob Friedman for references. Further topics: Work on homological, algebraic, and rational equivalence culminating in Clemens' paper on the non-finite generation of certain subquotients of chow groups (can't remember exactly which one).
6. Lecture on variations of Hodge structure
Speaker: Aise Johan de Jong.
Date: Friday, October 15.
Explain how a family of smooth projective varities over a smooth variety gives rise to a variation of Hodge structures. Define what is a variation of Hodge structures. Define the period map associated to a variation of Hodge structures. Give examples.
7. Lecture on the Hodge structure of a cubic threefold.
Speaker: Xuanyu Pan.
Date: Friday, October 22.
Following Clemens and Griffiths study the Hodge structure on a cubic threefold. This leads to a "simple" proof that a nonsingular cubic threefold is not rational. Really amazing stuff. See
• Clemens, C. Herbert; Griffiths, Phillip A. The intermediate Jacobian of the cubic threefold. Ann. of Math. (2) 95 (1972), 281--356.
Addendum: The most difficult part of the proof is to show that the intermediate Jacobian is not a Jacobian of a curve. I do not remember how that goes, but Bob Friedman suggests the following: "Use Prym varieties as in an argument by Mumford. There is a clear exposition by Beauville in some conference proceedings."
8. Lefschetz-Noether loci.
Speaker: Alexander Ellis.
Date: Friday, Octobre 29.
Here are some references:
• Green, Mark L. Components of maximal dimension in the Noether-Lefschetz locus. J. Differential Geom. 29 (1989), no. 2, 295--302.
• Voisin, Claire Une précision concernant le théorème de Noether. Math. Ann. 280 (1988), no. 4, 605--611.
• Voisin, Claire Composantes de petite codimension du lieu de Noether-Lefschetz. Comment. Math. Helv. 64 (1989), no. 4, 515--526. (Especially, look at Proposition 0.8 -- still not exactly right!)
• Voisin, Claire Sur le lieu de Noether-Lefschetz en degrés $6$ et $7$. Compositio Math. 75 (1990), no. 1, 47--68.
9. Deformations of Calabi-Yau's.
Speaker: Alex Waldron.
Date: Friday, November 5.
By a miracle these are unobstructed, so the local structure is clear. Less is known for sure about the global structure of the moduli space. Start with
• Gross, Mark Deforming Calabi-Yau threefolds. Math. Ann. 308 (1997), no. 2, 187--220.
10. Torelli for K3 surfaces.
Speaker: Bob Friedman.
Date: Friday, November 12.
It turns out that for K3-surfaces there is a Torelli theorem that is almost as good as Torelli for curves. We will have a guest lecture by Bob Friedman about this and a second lecture about applications. Ask Bob Friedman for references and about applications.
• Géométrie des surfaces K3: modules et périodes. A Beauville, JP Bourguignon, M Demazure - Astérisque 126, 1985
11. Picard numbers of surfaces in 3-dimensional Weighted Projective Spaces
Speaker: 漆 游
Date: Friday, November 19.
Notes of the talk.
As references take a look at the following:
• Cox, David A. Picard numbers of surfaces in 3-dimensional weighted projective spaces. Math. Z. 201 (1989), no. 2, 183--189.
• A.J. de Jong, J.H.M. Steenbrink, Picard numbers of surfaces in 3-dimensional weighted projective spaces, Mathematisches Zeitschrift, 206 (1991), pp. 341-344.
Let F be the family of quasi-smooth surfaces X in P(q_0,q_1, q_2,q_3) of degree km, m = lcm(q_0,q_1,q_2,q_3). Assume that m > 1 and that no three of q_0,q_1,q_2,q_3 have a common factor greater than one. Then the author shows that either p_g(X) = 0 or the generic member X of F has Picard number \rho(X) = 1. This is done by using methods of infinitesimal variation of Hodge structure.
12. Hodge structure of a cubic fourfold (Hassett).
Speaker: Howard Nuer
Date: Friday, December 3.
The Hodge structure of a cubic fourfold is very interesting. It is possibly related to the very interesting still open question of whether every cubic fourfold is rational or not.
• Hassett, Brendan Special cubic fourfolds Compositio Math. 120 (2000), no. 1, 1--23.
• Hassett, Brendan Some rational cubic fourfolds. J. Algebraic Geom. 8 (1999), no. 1, 103--114.
13. Complete intersection with middle picard number 1 defined over Q (Terasoma).
Speaker: Louis Garcia
Date: Friday, Decomber 10.
• Tomohide Terasoma, Complete intersections with middle picard number 1 defined over Q, Math. Z. 189 289--296 (1985).
• Davesh Maulik and Bjorn Poonen, NERON-SEVERI GROUPS UNDER SPECIALIZATION, Arxiv (recent preprint in AG)
14. (Generic) Torelli for hypersurfaces (2x).
Here we need the description of the cohomology of a hypersurface V(F) in terms of the Jacobian ideal of F, we need to learn about infinitesimal variations, etc. See for example
• Carlson, James A.; Griffiths, Phillip A. Infinitesimal variations of Hodge structure and the global Torelli problem Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 51--76, Sijthoff & Noordhoff, Alphen aan den Rijn---Germantown, Md., 1980.
• Donagi, Ron Generic Torelli for projective hypersurfaces Compositio Math. 50 (1983), no. 2-3, 325--353.
• Voisin, Claire Théorème de Torelli pour les cubiques de $P^5$. (French) [A Torelli theorem for cubics in $P^5$] Invent. Math. 86 (1986), no. 3, 577--601.
15. Some cases of the Hodge conjecture for 4-dimensional abelian varieties (Schoen, van Geemen).
There are some very limited cases where you can prove certain hodge classes on abelian fourfolds are algebraic. See
• Schoen, Chad Hodge classes on self-products of a variety with an automorphism. Compositio Math. 65 (1988), no. 1, 3--32.
• Schoen, Chad Addendum to: Hodge classes on self-products of a variety with an automorphism'' [Compositio Math. 65 (1988), no. 1, 3--32.] Compositio Math. 114 (1998), no. 3, 329--336.
• van Geemen, Bert An introduction to the Hodge conjecture for abelian varieties. Algebraic cycles and Hodge theory (Torino, 1993), 233--252, Lecture Notes in Math., 1594, Springer, Berlin, 1994.
• van Geemen, Bert Theta functions and cycles on some abelian fourfolds. Math. Z. 221 (1996), no. 4, 617--631.