[Paper Review] Some remarks on the group of derived autoequivalences

Fabrice Rosay|ArXiv.org|Jul 22, 2009

Algebraic Geometry and Number Theory11 references5 citations

TL;DR

This paper establishes that the neutral component of the derived autoequivalence group of a smooth projective variety over an algebraically closed field is isomorphic to the semi-direct product of the neutral component of its automorphism group and the neutral component of its Picard group. Using this structural result, the authors prove that there are at most countably many smooth projective varieties derived equivalent to a given variety, and they formulate a conjecture on boundedness of kernels in derived equivalences.

ABSTRACT

We prove that the neutral component of the group of derived autoequivalences of a smooth projective variety is the semi-direct product of the neutral component of its Picard group and its group of automorphisms. We use this result to prove several results concerning pairs of derived equivalent varieties.

Motivation & Objective

• To determine the structure of the neutral component of the group of triangulated autoequivalences of the bounded derived category of coherent sheaves on a smooth projective variety.
• To apply this structural result to understand the finiteness properties of derived equivalent varieties.
• To reformulate and provide evidence for a conjecture on the boundedness of Fourier-Mukai kernels in derived equivalences.
• To show that the number of smooth projective varieties derived equivalent to a given variety is at most countable.
• To establish a criterion for isomorphism of derived equivalent varieties based on the connected components of their derived autoequivalence groups.

Proposed method

• Construct an algebraic group scheme $\mathrm{Aut}^D_X$ parameterizing exact autoequivalences of $D(X)$ via integral transforms with perfect kernels.
• Use the composition law of integral transforms to define a group structure on the moduli space of such kernels.
• Apply Toën-Vaquié's framework to show $\mathrm{Aut}^D_X$ is a locally algebraic group scheme.
• Prove that the neutral component $ (\mathrm{Aut}^D_X)^0 $ is isomorphic to $ \mathrm{Aut}^\circ(X) \ltimes \mathrm{Pic}^\circ(X) $ via a decomposition of autoequivalences into automorphisms and line bundle twists.
• Use the boundedness criterion from [17] (Proposition 3.11) to relate cohomological bounds on kernels to quasi-compactness of families.
• Apply Lemma 3.6 (Favero) to show that if two varieties have isomorphic connected components of $\mathrm{Aut}^D$ containing twist functors by ample line bundles, then the varieties are isomorphic.

Experimental results

Research questions

• RQ1 What is the structure of the neutral component of the derived autoequivalence group of a smooth projective variety?
• RQ2 How many smooth projective varieties can be derived equivalent to a given variety?
• RQ3 Under what conditions can two derived equivalent varieties be shown to be isomorphic?
• RQ4 Can the kernels of Fourier-Mukai transforms between derived equivalent varieties be uniformly bounded in cohomological dimension?
• RQ5 Is there a conjectural criterion for finiteness of derived equivalence classes based on cohomological bounds on kernels?

Key findings

• The neutral component of the derived autoequivalence group $ \mathrm{Aut}^D(X)^0 $ is isomorphic to the semi-direct product $ \mathrm{Aut}^\circ(X) \ltimes \mathrm{Pic}^\circ(X) $, where the action is by pullback of line bundles.
• There are at most countably many smooth projective varieties (up to isomorphism) that are derived equivalent to a given smooth projective variety over an algebraically closed field.
• Two smooth projective varieties $ Y $ and $ Z $ are isomorphic if their derived autoequivalence groups have isomorphic connected components containing the twist functors by ample line bundles.
• The conjecture that the family of kernels $ \mathcal{K}_i^\cdot $ of derived equivalences is bounded (in the sense of cohomological bounds) is equivalent to a finiteness condition on derived equivalence classes.
• The boundedness of cohomological dimensions $ \dim \mathbb{H}^j(X, \mathcal{K}_i^\cdot \otimes \mathcal{L}^l) $ for $ l \in [0,d] $, with $ d \leq 2\dim X $, implies that the number of such kernels is finite, hence the number of derived equivalent varieties is finite up to isomorphism.
• The existence of a strong generator $ \mathcal{L}^\oplus \cdots \oplus \mathcal{L}^{-d} $ of $ D^b_{qc}(X \times X) $ with $ d \leq 2\dim X $ ensures the applicability of the boundedness criterion.