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In the paper « Matroids and the integral Hodge conjecture for abelian varieties », Philip Engel, Stefan Schreieder and myself prove that the cohomology class of any curve on a very general principally polarized abelian variety of dimension at least four is an even multiple of the minimal class. The same holds for the intermediate Jacobian of a very general cubic threefold. This disproves the integral Hodge conjecture for abelian varieties and shows that very general cubic threefolds are not stably rational. The goal of this lecture series is to explain the proofs of these results. Techniques related to monodromy, degenerations of abelian varieties over higher-dimensional bases, matroid theory, and birational geometry play a role, as I shall explain in the lectures. 

Lecture 1. Presentation of the main results: failure of the integral Hodge conjecture for abelian varieties and the existence of non stably rational cubic threefolds 

21 October, 13:30-14:30,HFG 7.07

In this first lecture, I will first provide an introduction to the general framework: the integral Hodge conjecture, which is a property that can hold or fail for smooth projective varieties over the complex numbers. Examples of varieties that fail that property were provided for the first time by Atiyah and Hirzebruch. Nonetheless, there are interesting classes of varieties that do satisfy the integral Hodge conjecture. I will also explain in this lecture the connection between the integral Hodge conjecture for abelian varieties and (stable) rationality of cubic threefolds, as discovered by Clemens—Griffiths and Voisin. I will present the two main theorems of our work, which imply that the integral Hodge conjecture for abelian varieties fails, and that not every smooth cubic threefold is stably rational. I will sketch an overview of the proof. 

Lecture 2. Degenerations of abelian varieties associated to regular matroids

28 October, 13:30-14:30, HFG 7.07

Mumford constructed degenerations of abelian varieties in his influential paper « An analytic construction of degenerating abelian varieties over complete rings ». In this second lecture, I will explain the main ideas behind the Mumford construction in the complex analytic setting, in which toric geometry plays a key role. I will also introduce the notion of matroid, and explain how to associate a degeneration of abelian varieties over a higher dimensional base to a regular matroid. This lecture will be based on the paper « Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction » by Engel, Schreieder and myself. 

Lecture 3. Spreading out the curve and reduction to combinatorics 

3 November,10:00-11:00, HFG 7.07 

The goal of this lecture is to reduce the failure of the integral Hodge conjecture for curve classes on principally polarized abelian varieties (resp. intermediate Jacobians of smooth cubic threefolds) to a combinatorial statement. The reduction step is provided in Theorem 1.8 of our paper « Matroids and the integral Hodge conjecture for abelian varieties ». In this lecture, we will assume the existence of a curve on a very general fiber of a matroidal family of principally polarized abelian varieties, whose cohomology class is a coprime-to-\ell multiple of the minimal class for some prime number \ell. The goal is to show that the corresponding regular matroid is then, in some sense, related to a graph. This reduction step is the geometric heart of the proof: among other things, we spread out the curve after a generically finite cover of the base, we construct a specific resolution of the resulting base changed total space of the degeneration of abelian varieties, and we replace the family of curves by a nice complete intersection in some projective bundle over the resolved total space, in order to achieve our goal.

Lecture 4. Finish the proof: quadratic splittings in cographic matroids 

10 November, 10:00-11:00, HFG 7.07

One way to construct matroids is from graphs, yielding the so-called classes of graphic and cographic matroids. After the reduction step in the previous lecture, it remains to prove Theorem 1.9 of our paper. This theorem is a combinatorial result, roughly saying that if a regular matroid is « close to a cographic matroid » in a suitable sense, then that matroid must in fact be cographic itself. The main theorems follow because there are non-cographic matroids, which we can use as input for our matroidal family of principally polarized abelian varieties.