On the Picard Varieties of Surfaces with Equivalent Derived Categories

It was shown recently by Popa and Schnell that the irregularities of two smooth projective varieties with equivalent bounded derived categories of coherent sheaves are equal. They conjectured that the Picard varieties of smooth projective varieties with equivalent derived categories are derived equivalent. This thesis investigates this conjecture for the case of smooth projective surfaces. More specifically, we showed that the Picard varieties of derived equivalent surfaces are in fact derived equivalent with the possible exception of the case of properly elliptic surfaces with constant $j$-invariant. In that case, we also provide an analysis of the Picard variety. In addition, we give a statement about the automorphism groups of derived equivalent surfaces.