Néron–Severi Lie Algebra, Autoequivalences of the Derived Category, and Monodromy

This is the authors' abstract. We don't add key points for this paper.

Let X be a smooth complex projective variety.The group of autoequivalences of the derived category of X acts naturally on its singular cohomology H • (X, Q)Zariski closure.We study the relation of the Lie algebra LieG eq (X) and the Neron-Severi Lie algebra gNS(X) ⊂ End(H(X, Q)) in case X has trivial canonical line bundle.At the same time for mirror symmetric families of (weakly) Calabi-Yau varieties we consider a conjecture of Kontsevich on the relation between the monodromy of one family and the group G eq (X) for a very general member X of the other family.