Séminaires : Séminaire de géométrie algébrique

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septembre-décembre Sophie-Germain, janvier-mars ENS, avril-juin Jussieu

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Résumé: Every abelian variety is a quotient of a Jacobian, but to quantify that seems very difficult: Given an abelian variety A of dimension g over a field K, what is the smallest dimension C_K(g) such that A is a quotient of a Jacobian of dimension C_K(g) (Or equivalently, admits a map from a smooth curve of that genus).

This question is extremely difficult even over C, where we have polynomial lower bounds and super-exponential upper bounds on C_K(g). Over a countable field like Qbar things become even more difficult. Conjecturally, one would expect that C_K(g) should be the same for K=C or K=Qbar, but even showing that there are abelian varieties over Qbar not isogenous to Jacobians  (i.e. that C_{Qbar}(g)>g) was unknown for a long time. We present a proof that C_{Qbar}(g)>=2g (for g>=5, C_{Qbar}(4)=7) . We explain how to interpret this question in the framework of unlikely-intersections, and the ideas that go into the proof. The proof follows the by-now familiar Pila-Zannier method, using the latest results on the Zilber-Pink conjecture, and the main new ingredient is a new lower bound on certain Galois orbits.