### Chương trình

Thứ 7, ngày 25 tháng 8 năm 2012

- 13h30-14h15: Tea

- 14h15-15h45:  Bài giảng “On the arithmetic of hyperelliptic curves” của GS. Benedict Gross.

- 15h45-16h00: Tea

- 16h00-17h30: Bài giảng “Automorphism groups of compact Kaehler manifolds” của GS. Đinh Tiến Cường

Chủ nhật, ngày 26 tháng 8 năm 2012

Sáng:

- 8h30-8h45: Tea

- 8h45-10h15: Bài giảng “Finiteness and companions, after P. Deligne and V. Drinfeld” của GS. Hélène Esnault.

- 10h15-10h30: Tea

- 10h30-12h00: Bài giảng “Realizing unstable modules as the cohomology of spaces, a survey” của GS. Lionel Schwartz.

Chiều:

- 14h00-14h15: Tea

- 14h15-15h45: Bài giảng “Hyperbolic algebraic varieties and holomorphic differential equations của GS. Jean-Pierre Demailly.

- 15h45-16h00: Tea

- 16h00: Thảo luận về Hội thảo hàng năm của năm tiếp theo.

Nội dung các bài giảng:

Jean-Pierre Demailly: Hyperbolic algebraic varieties and holomorphic differential equations
Every complex space carries an invariant metric known as the Kobayashi metric, computed from extremal holomorphic curves. On projective varieties, Kobayashi hyperbolicity is equivalent to the non existence of entire holomorphic curves. In general it is expected that the existence of infinitely many rational points is related to the locus where entire holomorphic curves accumulate. An important conjecture due to Green-Griffiths and Lang asserts that for every projective algebraic variety of general type, there is a proper algebraic subvariety containing all entire curves, that would contain also all rational points but finitely many. The description of this locus is strongly related to certain global algebraic differential equations, given e.g. by holomorphic foliations. The geometry of jet bundles and their cohomology plays here an important role. We will discuss recent progress on these questions.

Hélène Esnault: Finiteness and companions, after P. Deligne and V. Drinfeld

Goal will be to present Deligne’s existence of a number field receiving the coefficients of an $\ell$-adic lisse sheaf of weight 0 on a normal scheme $X$ over a finite field, Drinfeld’s existence of $\ell’$-adic companions on $X$ lisse (Deligne’s conjecture1.2.10 in Weil II), and Deligne’s finiteness result for the number of $\ell$-adic lisse sheaves to bounded rank and ramification.

Benedict Gross: On the arithmetic of hyperelliptic curves

In this talk, he will show how one can use invariant theory to study the arithmetic of hyperelliptic curves of a fixed genus over Q with a rational Weierstrass point. He will obtain an upper bound on the average rank of their Jacobians and will use this bound to restrict the number of rational points on the curve (when the genus is at least 2). This is a report on joint work with Manjul Bhargava.

Dinh Tien Cuong, Automorphism groups of compact Kaehler manifolds

Automorphisms of compact Kaehler manifolds are currently studied mostly from two points of view: Complex Dynamics and Algebraic Geometry. In this talk, we will survey some classical results, e.g. Theorems by Lieberman, Gromov, Yomdin,

Lionel Schwartz: Realizing unstable modules as the cohomology of spaces, a survey

The talk will describe recent results and conjectures about the following question: when an unstable module can be the mod p cohomology of a space. One will focuss on qualitative results (but not only) rather than on specific cases.