On derived equivalence classes of algebraic varieties

Abstract

Let $X\to S$ be a miniversal family of smooth and projective varieties and D be a fixed triangulated category. We show that the set of points s in S such that the derived category of the fiber X_s at s is equivalent to D is at most countable. We deduce from this that the derived equivalence classes of smooth and projective complex varieties is at most countable.

19 pages, French, some typos corrected. Final version, to appear in JAG