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We introduce a surgery operation on symplectic manifolds called coisotropicLuttinger surgery, which generalizes Luttinger surgery on Lagrangian tori insymplectic 4-manifolds. We use it to produce infinitely many distinctsymplectic non-Kahler 6-manifolds $X$ with $c_1(X)=0$ which are not of the form$M\times F$ for $M$ a symplectic 4-manifold and $F$ a closed surface.

Coisotropic Luttinger surgery and some new symplectic 6-manifolds with vanishing canonical class