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]] | + | 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 16:02, 22 January 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
The Milnor hypersurfaces, denoted , are a family of smooth manifolds that generate (with redundancy) the complex bordism ring.
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.
3 Invariants
The signature of the Milnor hypersurfaces is known:
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==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.
3 Invariants
The signature of the Milnor hypersurfaces is known: