Milnor Hypersurfaces

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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.
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 [[Wikipedia:Segre_embedding|Segre embedding]] $$\CP^i \times \CP^j$\to\CP^{i+j-1}$$ transversely.
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This equation defines a generic hyperplane intersecting the image of the [[Wikipedia:Segre_embedding|Segre embedding]] $$\CP^i \times \CP^j \to \CP^{i+j-1}$$ transversely.
The role of these manifolds in complex bordism is described on the page [[Complex bordism#Milnor hypersurfaces|Complex bordism]].
The role of these manifolds in complex bordism is described on the page [[Complex bordism#Milnor hypersurfaces|Complex bordism]].

Revision as of 10:01, 18 January 2013

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

Contents

1 Introduction

The Milnor hypersurfaces, denoted 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, 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
\displaystyle \CP^i \times \CP^j \to \CP^{i+j-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}

Proof. Click here - opens a separate pdf file.

\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]] $$\CP^i \times \CP^j$\to\CP^{i+j-1}$$ 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, 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
\displaystyle \CP^i \times \CP^j \to \CP^{i+j-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}

Proof. Click here - opens a separate pdf file.

\square


4 References

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