Derived equivalences of K3 surfaces and orientation
Daniel Huybrechts, Emanuele Macri, Paolo Stellari
Number | 22 |
---|---|
Authors |
Emanuele Macrì
Daniel Huybrechts |
Year | 2007 |
Every Fourier--Mukai equivalence between the derived categories of two K3
surfaces induces a Hodge isometry of their cohomologies viewed as Hodge
structures of weight two endowed with the Mukai pairing. We prove that this
Hodge isometry preserves the natural orientation of the four positive
directions. This leads to a complete description of the action of the group of
all autoequivalences on cohomology very much like the classical Torelli theorem
for K3 surfaces determining all Hodge isometries that are induced by
automorphisms.
The appendix contains a description of Lieblich's obstruction for the
deformation of complexes in families in terms of Kodaira--Spencer and Atiyah
classes which is of independent interest.