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# Homology and Cohomology of Toric Varieties

Type of Publication: | Preprint |

URI (citable link): | http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-20372 |

Author: | Jordan, Arno |

Year of publication: | 1998 |

Series: | Konstanzer Schriften in Mathematik und Informatik ; 57 |

Summary: |
By the Theorem of Jurkiewicz-Danilov, there is an isomorphism of graded rings between the integral cohomology ring and the Chow ring of a "smooth" compact toric variety; moreover, an explicit computation of both rings is provided with the help of the finitely many integral data encoded in the fan that defines the toric variety. As a generalization to even noncompact toric varieties with arbitrary singularities, we show the following:
There is a spectral sequence - induced by the orbit structure - that converges to the integral (resp. rational) homology with closed supports of such a toric variety, where the Chow groups appear as the diagonal E2-terms. In addition, for homology with closed supports and coefficients in an abelian group, the homology groups in low and in high degrees of any toric variety are explicitly computed. Finally, we develop the dual theory for cohomology with compact supports and coefficients in an abelian group, generalizing Fischli's approach and results for the computation of the integral cohomology of compact toric varieties. |

Subject (DDC): | 004 Computer Science |

Comment on publication: | Buchhandelsausg. erschienen 1998 im Verl. Hartung-Gorre, Konstanz. ISBN 3-89649-273-X |

Link to License: | In Copyright |

Checksum:
MD5:f5f5b6f4d5512110a711e13c8534d972

JORDAN, Arno, 1998. Homology and Cohomology of Toric Varieties

@unpublished{Jordan1998Homol-6176, title={Homology and Cohomology of Toric Varieties}, year={1998}, author={Jordan, Arno}, note={Buchhandelsausg. erschienen 1998 im Verl. Hartung-Gorre, Konstanz. ISBN 3-89649-273-X} }

preprint_057.pdf | 305 |