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We study the symplectic topology of certain K3 surfaces (including the"mirror quartic" and "mirror double plane"), equipped with certain K\"ahlerforms. In particular, we prove that the symplectic Torelli group may beinfinitely generated, and derive new constraints on Lagrangian tori. The keyinput, via homological mirror symmetry, is a result of Bayer and Bridgeland onthe autoequivalence group of the derived category of an algebraic K3 surface ofPicard rank one.

Symplectic topology of K3 surfaces via mirror symmetry