## Signatures of singular branched covers.(English)Zbl 0793.57013

We consider branched cyclic $$d$$-fold covers of a closed smooth oriented manifold $$M$$ along a collection of codimension-2 smooth oriented submanifolds $$\{K_ i\}$$ which are in general position, where the branching index of each submanifold is $$d$$. It is shown that these are rational homology manifolds. A formula for the eigenspace signatures of the cover in terms of the Hirzebruch $$L$$-class of $$M$$ and 2-dimensional cohomology classes dual to the $$K_ i$$ is given. Many singular algebraic varieties arise in this manner. As an application the signature of a $$d$$- fold branched cyclic cover of $$\mathbb{C} \text{P} (2k)$$ along a collection of $$d$$ hyperplanes in general position is calculated to be $$1+{d-1 \choose k+1}$$. A formula for the stable tangent bundle of a branched cover along a single submanifold is also given.

### MSC:

 57R20 Characteristic classes and numbers in differential topology 14F45 Topological properties in algebraic geometry 57M12 Low-dimensional topology of special (e.g., branched) coverings
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### References:

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