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Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology


Authors: Luis A. Cordero, Marisa Fernández, Alfred Gray and Luis Ugarte
Journal: Trans. Amer. Math. Soc. 352 (2000), 5405-5433
MSC (2000): Primary 32L05, 55P62, 57T15; Secondary 53C56, 32Q99
DOI: https://doi.org/10.1090/S0002-9947-00-02486-7
Published electronically: June 28, 2000
MathSciNet review: 1665327
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Abstract: We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if $\Gamma \backslash G$ is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}(\Gamma \backslash G)$ is canonically isomorphic to the ${\bar{\partial}}$-cohomology $H^{\ast ,\ast }_{{\bar{\partial}}}({\mathfrak g}^{{\mathbb C}})$of the bigraded complex $(\Lambda ^{\ast ,\ast } ({\mathfrak g}^{{\mathbb C}})^{\ast }, {\bar{\partial}})$of complex valued left invariant differential forms on the nilpotent Lie group $G$.


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Additional Information

Luis A. Cordero
Affiliation: Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15705 Santiago de Compostela, Spain
Email: cordero@zmat.usc.es

Marisa Fernández
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
Email: mtpferol@lg.ehu.es

Alfred Gray
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: gray@bianchi.umd.edu

Luis Ugarte
Affiliation: Departamento de Matemáticas (Geometría y Topología), Facultad de Ciencias, Universidad de Zaragoza, Campus Plaza San Francisco, 50009 Zaragoza, Spain
Email: ugarte@posta.unizar.es

DOI: https://doi.org/10.1090/S0002-9947-00-02486-7
Received by editor(s): March 29, 1998
Received by editor(s) in revised form: December 20, 1998
Published electronically: June 28, 2000
Additional Notes: Partially supported by Xunta de Galicia (Spain), Research Project XUGA20701B93, U.P.V. Project 127.310–EA 191/94, DGICYT Projects PB94-0633-C02-01 and PB94-0633-C02-02, and by DGICYT Sabbatical Program 1995-0735.
The revised version was submitted just after the unexpected death of A. Gray in Bilbao (Spain) on October 27, 1998.
Article copyright: © Copyright 2000 American Mathematical Society

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