  We prove that multiplicative preprojective algebras, defined by
Crawley-Boevey and Shaw, are 2-Calabi-Yau algebras, in the case of quivers
containing unoriented cycles. If the quiver is not itself a cycle, we show that
the center is trivial, and hence the Calabi-Yau structure is unique. If the
quiver is a cycle, we show that the algebra is a non-commutative crepant
resolution of its center, the ring of functions on the corresponding
multiplicative quiver variety with a type A surface singularity. We also prove
that the dg versions of these algebras (arising as certain Fukaya categories)
are formal. We conjecture that the same properties hold for all non-Dynkin
quivers, with respect to any extended Dynkin subquiver (note that the cycle is
the type A case). Finally, we prove that multiplicative quiver varieties-for
all quivers-are formally locally isomorphic to ordinary quiver varieties. In
particular, they are all symplectic singularities (which implies they are
normal and have rational Gorenstein singularities). This includes character
varieties of Riemann surfaces with punctures and monodromy conditions. We
deduce this from a more general statement about 2-Calabi--Yau algebras
(following Bocklandt, Galluzzi, and Vaccarino).
