Milnor Hypersurfaces
From Manifold Atlas
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== Introduction == | == Introduction == | ||
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== Construction and examples == | == Construction and examples == | ||
<wikitex>; | <wikitex>; | ||
− | For fixed natural numbers $i \ | + | 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]] $\CP^i \times \CP^j \to \CP^{(i+1)(j+1)-1}$ transversely. | ||
− | The role of these manifolds in complex bordism is | + | The role of these manifolds in complex bordism is described on the page [[Complex bordism#Milnor hypersurfaces|Complex bordism]]. |
</wikitex> | </wikitex> | ||
− | == Invariants == | + | |
+ | ==Invariants== | ||
<wikitex>; | <wikitex>; | ||
− | .. | + | 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}} | ||
+ | |||
</wikitex> | </wikitex> | ||
Latest revision as of 16:04, 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 , are a family of smooth manifolds that generate (with redundancy) the complex bordism ring.
[edit] 2 Construction and examples
For fixed natural numbers , is defined as the hypersurface in satisfying the equation , where and are homogeneous coordinates for and respectively. This equation defines a generic hyperplane intersecting the image of the Segre embedding 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: