Eseen 2010 5th day, Algebraic Geometry and Arithmetic

2010-Feb-20th was the final day of the official day of Essen 2010, whose schedule was adapted to the carnival that I mentioned previously.

Thelene (CNRS & Paris) and Lazarsfield were the speakers of this day. Thelene talked about the universal invariant and algebraic cycles of higher codimension. The main result was

Z^{2i} (X) = H^{2i}_{Hodge} (X, ¥mathbb{Z}) / H^{2i}_{alg} (X, ¥mathbb{Z})

(where i = 1: Lefshectz, d-1, or 2). He considered Bloch-Ogus theorem and Betti cohomology.

Lazarsfield talked about the positivity (Eckart Viehweg and Esnault) of cycles on abelian varieties. I. Review of positivity for divisors of X = smooth projective variety / ¥mathbb{C}.

-- 1960's (Kodaira and Kleiman) for numerical theory of positivity.

II. Higher codimension:

-- product structure on the nef class (after the letter of Grothendieck to Mumford in the collection Vol. II.), especially on whether nef class is in the pseudo-effective divisors.

III. Positivity of Abelian varieties of real (k,k)-forms on V, where we set B = V / ¥Lambda for abelian variety. He stated the notion of strong / weak positivity of (k,k)- differential form.

After the conference, some of the participants are guided to the coal mine called Zeche Zollverein, which was about 20 minutes far from the Berliner Platz station by "tram".

I could have talked to famous professors as well as young fellows in this week. Therefore I appreciated this occasion of conference on the memory of Prof. Eckard Viehweg.