On rigidity of complex Hirzebruch genera on $SU$-manifolds
We prove that if a complex genus $φ\colon \varOmega^U \to R$ is rigid on $SU$-manifolds with a torus action then $φ$ is the elliptic Krichever genus.
math.AT↗
arXiv subjects
Georgy Chernykh
Publications and source records attributed to Georgy Chernykh.
$SU$-linear operations in complex cobordism and the $c_1$-spherical bordism theory
We study the $SU$-linear operations in complex cobordism and prove that they are generated by the well-known geometric operations $\partial_i$. For the theory $W$ of $c_1$-spherical bordism, we describe all $SU$-linear multiplications on $W$ and projections $MU \to W$. We also analyse complex orientations on $W$ and the corresponding formal group laws $F_W$. The relationship between the formal group laws $F_W$ and the coefficient ring $\varOmega^W$ of the $W$-theory was studied by Buchstaber in 1972. We extend his results by showing that for any $SU$-linear multiplication and orientation on $W$, the coefficients of the corresponding formal group law $F_W$ do not generate the ring $\varOmega^W$, unlike the situation with complex bordism.
math.AT↗
Landweber exactness of the formal group law in $c_1$-spherical bordism
We describe the structure of the coefficient ring $W^*(pt)=\varOmega_W^*$ of the $c_1$-spherical bordism theory for an arbitrary $SU$-bilinear multiplication. We prove that for any $SU$-bilinear multiplication the formal group of the theory $W^*$ is Landweber exact. Also we show that after inverting the set $\mathcal P$ of Fermat primes there exists a complex orientation of the localized theory $W^*[\mathcal P^{-1}]$ such that the coefficients of the corresponding formal group law generate the whole coefficient ring $\varOmega_W^*[\mathcal P^{-1}]$.
math.AT↗
SU-bordism: structure results and geometric representatives
In the first part of this survey we give a modernised exposition of the structure of the special unitary bordism ring, by combining the classical geometric methods of Conner-Floyd, Wall and Stong with the Adams-Novikov spectral sequence and formal group law techniques that emerged after the fundamental 1967 work of Novikov. In the second part we use toric topology to describe geometric representatives in SU-bordism classes, including toric, quasitoric and Calabi-Yau manifolds.
math.AT↗