Milnor Hypersurfaces

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This page has not been refereed. The information given here might be incomplete or provisional.


[edit] 1 Introduction

The Milnor hypersurfaces, denoted H_{ij}, are a family of smooth manifolds that generate (with redundancy) the complex bordism ring.

[edit] 2 Construction and examples

For fixed natural numbers 0 \leq i \leq j, H_{ij} is defined as the hypersurface in \CP^i \times \CP^j satisfying the equation x_0z_0 + ... + x_iz_i = 0, where x_k and z_k are homogeneous coordinates for \CP^i and \CP^j respectively. This equation defines a generic hyperplane intersecting the image of the Segre embedding \CP^i \times \CP^j \to \CP^{(i+1)(j+1)-1} transversely.

The role of these manifolds in complex bordism is described on the page Complex bordism.

[edit] 3 Invariants

The signature of the Milnor hypersurfaces is known:

Proposition 3.1.

\displaystyle  \sigma(H_{ij})=\begin{cases}     1 & i \textrm{ even, } j \textrm{ odd} \\     0 & \textrm{otherwise} \end{cases}

Proof. Click here - opens a separate pdf file.


[edit] 4 References

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