Milnor Hypersurfaces

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== Introduction ==
== Introduction ==
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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.
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This equation defines a generic hyperplane intersecting the image of the [[Wikipedia:Segre_embedding|Segre embedding]] $\CP^i \times \CP^j \to \CP^{(i+1)(j+1)-1}$ transversely.
The role of these manifolds in complex bordism is descbribed on the page [[Complex bordism#Milnor hypersurfaces|Complex bordism]].
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The role of these manifolds in complex bordism is described on the page [[Complex bordism#Milnor hypersurfaces|Complex bordism]].
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== Invariants ==
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==Invariants==
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...
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The signature of the Milnor hypersurfaces is known:
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{{beginthm|Proposition}} \label{prop:signature-milnor}
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$$
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\sigma(H_{ij})=\begin{cases}
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1 & i \textrm{ even, } j \textrm{ odd} \\
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0 & \textrm{otherwise}
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\end{cases}
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$$
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{{beginproof}}
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[[Media:signature-milnor.pdf|Click here - opens a separate pdf file]].
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{{endproof}}
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Latest revision as of 15:04, 22 January 2013

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

Contents

[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.

\square


[edit] 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. The role of these manifolds in complex bordism is descbribed on the page [[Complex bordism#Milnor hypersurfaces|Complex bordism]]. == Invariants == ; ... == References == {{#RefList:}} [[Category:Manifolds]]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.

\square


[edit] 4 References

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