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We study the $SU$-linear operations in complex cobordism and prove that theyare generated by the well-known geometric operations $\partial_i$. For thetheory $W$ of $c_1$-spherical bordism, we describe all $SU$-linearmultiplications on $W$ and projections $MU \to W$. We also analyse complexorientations on $W$ and the corresponding formal group laws $F_W$. Therelationship between the formal group laws $F_W$ and the coefficient ring$\varOmega^W$ of the $W$-theory was studied by Buchstaber in 1972. We extendhis results by showing that for any $SU$-linear multiplication and orientationon $W$, the coefficients of the corresponding formal group law $F_W$ do notgenerate the ring $\varOmega^W$, unlike the situation with complex bordism.

$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory