On signatures associated with ramified coverings and embedding problems

Wood, J.; Thomas, Emery

Annales de l'Institut Fourier, Tome 23 (1973) no. 2, pp. 229-235.

Étant donné une classe de cohomologie $\xi \in H^2(M;Z)$, il existe une sous-variété $K\subset M$ duale à $\xi$ dans le sens de Poincaré. Il existe un ensemble de tels plongements qui est caractérisé par des propriétés topologiques, que les hypersurfaces algébriques de $\mathbf{C}{P}_n$ vérifient. Cet exposé résume quelques résultats sur la question : comment la divisibilité de $\xi$ limite-t-elle les sous-variétés duales, $K$, dans cet ensemble ? Et nous donnons une formule pour la signature associée à un revêtement d’ordre $d$ sur $M$, ramifiée sur $K$ ; nous le démontrons dans le cas où $d=2$.

Given a cohomology class $\xi \in H^2(M;Z)$ there is a smooth submanifold $K\subset M$ Poincaré dual to $\xi$. A special class of such embeddings is characterized by topological properties which hold for nonsingular algebraic hypersurfaces in $\mathbf{C}{P}_n$. This note summarizes some results on the question: how does the divisibility of $\xi$ restrict the dual submanifolds $K$ in this class ? A formula for signatures associated with a $d$-fold ramified cover of $M$ branched along $K$ is given and a proof is included in case $d=2$.

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