TL;DR
This paper characterizes when Hodge isometries of twisted K3 surfaces come from derived autoequivalences, describing the image of the autoequivalence representation within the group of all Hodge isometries.
Contribution
It provides a criterion for when a Hodge isometry arises from an autoequivalence of a twisted K3 surface and describes the index of the image subgroup.
Findings
The image of the autoequivalence representation has index one or two in the group of all Hodge isometries.
Both index cases (one and two) can occur for twisted K3 surfaces.
A criterion is established to determine when a Hodge isometry is induced by an autoequivalence.
Abstract
Derived equivalences of twisted K3 surfaces induce twisted Hodge isometries between them; that is, isomorphisms of their cohomologies which respect certain natural lattice structures and Hodge structures. We prove a criterion for when a given Hodge isometry arises in this way. In particular, we describe the image of the representation which associates to any autoequivalence of a twisted K3 surface its realization in cohomology: this image is a subgroup of index one or two in the group of all Hodge isometries of the twisted K3 surface. We show that both indices can occur.
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