Moscow Mathematical Journal

Volume 22, Issue 4, October–December 2022 pp. 705–739.

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

Authors: Valery A. Lunts (1)
Author institution:(1) Department of Mathematics, Indiana University, Bloomington, IN 47405,
National Research University Higher School of Economics, Moscow, Russia

Summary:

Let $X$ be a smooth complex projective variety. The group of autoequivalences of the derived category of $X$ acts naturally on its singular cohomology $H^\bullet (X,\mathbb{Q})$ and we denote by $G^{\mathrm{eq}}(X)\subset \mathrm{GL}(H^\bullet (X,\mathbb{Q}))$ its image. Let $\overline{G^{\mathrm{eq}}(X)}\subset \mathrm{GL}(H^\bullet (X,\mathbb{Q})$ be its Zariski closure. We study the relation of the Lie algebra $\mathrm{Lie}\, \overline{G^{\mathrm{eq}}(X)}$ and the Néron–Severi Lie algebra $\mathfrak{g}_{\mathrm{NS}}(X)\subset \mathrm{End}(H(X,\mathbb{Q}))$ 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 $G^{\mathrm{eq}}(X)$ for a very general member $X$ of the other family.

2020 Math. Subj. Class. 18G80, 14F08.

Keywords: Calabi–Yau varieties, derived categories, Néron–Severi Lie algebra, monodromy group.

Contents Full-Text PDF