Geodesic
Obstructions to the existence of $S^1$-actions. Bordism of ramified coverings
Izvestiya. Mathematics , Tome 10 (1976) no. 4, pp. 783-797

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In this article the author proves that the values of the multiplicative genera $A_k$ under discussion, where $k=2,3,\dots$, are obstructions to the existence of nontrivial $S^1$-actions on a unitary manifold whose first Chern class is divisible by $k$. The effective computation of these obstructions is carried out for algebraic manifolds. Simultaneously, formulas for the bordism class of a ramified covering are obtained. Bibliography: 8 titles. 
@article{IM2_1976_10_4_a6,
     author = {I. M. Krichever},
     title = {Obstructions to the existence of $S^1$-actions. {Bordism} of ramified coverings},
     journal = {Izvestiya. Mathematics },
     pages = {783--797},
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     volume = {10},
     number = {4},
     year = {1976},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1976_10_4_a6/}
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I. M. Krichever. Obstructions to the existence of $S^1$-actions. Bordism of ramified coverings. Izvestiya. Mathematics , Tome 10 (1976) no. 4, pp. 783-797. http://geodesic.mathdoc.fr/item/IM2_1976_10_4_a6/