Neron-Severi Lie algebra, autoequivalences of the derived category, and monodromy

Abstract

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) and we denote by Geq(X) the Zariski closure of its image in Gl(H(X, Q)). We study the relation of the Lie algebra LieGeq(X) and the Neron-Severi Lie algebra gNS(X) 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 Geq(X) for a very general member X of the other family.

This is the final version to appear in MMJ. Some mistakes have been corrected. In section 3 we removed a previously claimed theorem, whose proof was incorrect. In section 4 we now consider only the families of pseudo-ample lattice polarized K3 surfaces. Finally in section 5 we fixed the definition of the Hodge group. Also many clarifying comments we added