Derived and abelian equivalence of K3 surfaces

Abstract

The paper attempts to shed more light on a particular class of stability conditions on K 3 K3 surfaces constructed by Tom Bridgeland. The hearts of the underlying t-structures turn out to be significant invariants of the surface. We prove that two K 3 K3 surfaces X X and X ′ X’ are derived equivalent if and only if there exist complexified polarizations B + i ω B+i\omega and B ′ + i ω ′ B’+i\omega ’ such that the associated abelian categories A ( exp ⁡ ( B + i ω ) ) \mathcal {A}(\exp (B+i\omega )) and K ( exp ⁡ ( B ′ + i ω ′ ) ) \mathcal {K}(\exp (B’+i\omega ’)) are equivalent. We study in detail the minimal objects of A ( exp ⁡ ( B + i ω ) ) \mathcal {A}(\exp (B+i\omega )) and investigate stability under the Fourier–Mukai transform.