One of the most important problems in complex algebraic geometry is to understand the interaction between the topological, analytic, and algebraic properties of the solutions of a system of polynomial equations defined over the complex numbers.
The first pioneering step in this direction, due to Hodge in the 1930s, is the Hodge decomposition, which asserts that there exists a natural isomorphism \[ H^n(X,\mathbb Q)\otimes \mathbb C \simeq \bigoplus_{p+q=n} H^p(X,\Omega^q), \] where $H^n(X,\mathbb Q)$ denotes the singular cohomology of the smooth projective variety $X$ (a topological invariant), and $\Omega^q$ is its sheaf of holomorphic $q$-forms. This decomposition is obtained by passing through harmonic forms, which are analytic in nature.
On the algebraic side, for every $i \ge 0$, one has the Chow groups $\mathrm{CH}^i(X)$ of algebraic cycles of codimension $i$, consisting of rational linear combinations of algebraic subvarieties of $X$ of codimension $i$, modulo an appropriate equivalence relation.
Taking the fundamental class of a subvariety in cohomology yields the cycle class map \[ \mathrm{ch}^n : \mathrm{CH}^n(X) \longrightarrow H^{2n}(X,\mathbb Q), \] which relates algebraic cycles to topology. Describing its image is, in many respects, one of the most important open problems in complex algebraic geometry. In the 1950s, Hodge formulated what has become the famous Hodge conjecture: \[ \mathrm{Image}(\mathrm{ch}^n) = H^n(X,\mathbb Q)\cap H^n(X,\Omega^n), \] where the intersection is taken inside $H^n(X,\mathbb Q)\otimes \mathbb C$.
Such a result would establish a deep link between topology, analysis, and algebraic geometry. Aside from the case $n=1$, now known as the Lefschetz (1,1) theorem, and a few other sporadic results, very little is known about this conjecture for $n>1$.
The goal of this meeting is to study Eyal Markman’s recent proof of the Hodge conjecture for abelian fourfolds, a major result, and arguably the most significant advance on this conjecture since its formulation. Our discussions will provide an opportunity to become familiar with the central geometric techniques underlying Markman’s argument: the Hodge–Weil classes, the deformations of twisted sheaves, and the semi-regularity theorems.
We plan to revive the tradition of an annual series of conferences in arithmetic geometry based on the model of Oberwolfach's Arbeitsgemeinschaft. They were previously organized by Jean-Benoît Bost and François Loeser from 1995 to 2002 at Luminy. The first editions covered topics such as Euler systems, higher class field theory or modular forms and Galois representations.
The public we have in mind consists mainly of PhD students and early postdocs, with the aim of offering a friendly ambience to learn mathematical subjects that do not necessarily belong to one's own research area. At the end of each edition, the topic for the next one will be voted, and a scientific program will be written in close collaboration with a leading expert. About six months before the conference, participants will apply and the talks will be distributed among them.
Scientific Committee
Eyal Markman (University of Massachusetts)