Generators of the Algebraic Symplectic Bordism Ring

Algebraic symplectic cobordism is the universal symplectically oriented cohomology theory for schemes, represented by the motivic commutative ring spectrum MSp constructed by Panin and Walter. The graded algebraic diagonal MSp* of the coefficient ring of MSp is unknown. Through a symplectic version of the Pontryagin-Thom construction, one can associate any symplectic variety X with a symplectic class [X]MSp in MSp-dimX. Still, the problem in using these classes to study the ring MSp* is the paucity of non-trivial examples of symplectic varieties. We modify this construction to obtain elements in MSp* from a large family of varieties that are not symplectic but carry a certain "symplectic twist". Then, using a strategy relying on the Adams spectral sequence for MSp, we find a criterion to select generators among these classes, after taking a completion along the motivic Hopf map.