# Milnor Hypersurfaces

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

## 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$

## 4 References

\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$respectively. This equation defines a generic hyperplane intersecting the image of the [[Wikipedia:Segre_embedding|Segre embedding]]$\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#Milnor hypersurfaces|Complex bordism]]. ==Invariants== ; The signature of the Milnor hypersurfaces is known: {{beginthm|Proposition}} \label{prop:signature-milnor} $$\sigma(H_{ij})=\begin{cases} 1 & i \textrm{ even, } j \textrm{ odd} \ 0 & \textrm{otherwise} \end{cases}$$ {{beginproof}} [[Media:signature-milnor.pdf|Click here - opens a separate pdf file]]. {{endproof}} == References == {{#RefList:}} [[Category:Manifolds]]H_{ij}, 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$