# Milnor Hypersurfaces

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## 1 Introduction

The Milnor hypersurfaces, denoted $H_{ij}$${{Stub}} == Introduction == ; The Milnor hypersurfaces, denoted H_{ij}, are a family of smooth manifolds that generate (with redundancy) the [[Complex bordism|complex bordism]] ring. == Construction and examples == ; For fixed natural numbers , are a family of smooth manifolds that generate (with redundancy) the complex bordism ring.

## 2 Construction and examples

For fixed natural numbers $0 \leq i \leq j$$0 \leq i \leq j$, $H_{ij}$$H_{ij}$ is defined as the hypersurface in $\CP^i \times \CP^j$$\CP^i \times \CP^j$ satisfying the equation $x_0z_0 + ... + x_iz_i = 0$$x_0z_0 + ... + x_iz_i = 0$, where $x_k$$x_k$ and $z_k$$z_k$ are homogeneous coordinates for $\CP^i$$\CP^i$ and $\CP^j$$\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}$$\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.

## 3 Invariants

The signature of the Milnor hypersurfaces is known:

Proposition 3.1. $\displaystyle \sigma(H_{ij})=\begin{cases} 1 & i \textrm{ even, } j \textrm{ odd} \\ 0 & \textrm{otherwise} \end{cases}$ $\square$$\square$