A quotient of the stability manifold by an equivalence relation is a real manifold of half dimension, and the original manifold can be reconstructed from the quotient together with the relation ≲.
Derived equivalences of K3 surfaces and orientation
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abstract
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.
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A Real Reduction of the Manifold of Bridgeland Stability Conditions
A quotient of the stability manifold by an equivalence relation is a real manifold of half dimension, and the original manifold can be reconstructed from the quotient together with the relation ≲.