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

TL;DR

This paper explores the relationship between autoequivalences of derived categories, Neron-Severi Lie algebras, and monodromy in Calabi-Yau varieties, linking algebraic and geometric symmetries through mirror symmetry.

Contribution

It establishes connections between the Lie algebra of autoequivalences and Neron-Severi Lie algebra for varieties with trivial canonical bundle, and examines a conjecture relating monodromy and autoequivalence groups in mirror symmetry.

Findings

Relation between LieGeq(X) and gNS(X) for Calabi-Yau varieties

Evidence supporting Kontsevich's conjecture on monodromy and autoequivalences

Insights into symmetries in mirror symmetric families

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.