Lefschetz Complex Conditions for Complex Manifolds |
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Authors: | Cordero Luis A Fernández Marisa Ugarte Luis |
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Institution: | (1) Departamento de Geometría y Topología, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15706 Santiago de Compostela, Spain;(2) Departamento de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain;(3) Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, Campus Plaza San Francisco, 50009 Zaragoza, Spain |
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Abstract: | For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L
1,0: H
1,0(M) H
{n, n–1(M) and L
0, 1: H
0, 1(M) H
n – 1, n
(M) given by the cup product with ]
n – 1, n being the complex dimension of M andH
*, *(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L
1, 0 (resp.L
0, 1) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin Topology
27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L
1, 0characterizes those complex structures which are Abelian in the sense ofBarberis et al. Ann. Global Anal. Geom.
13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1). |
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Keywords: | complex manifold Dolbeault cohomology Abelian complex structure complex parallelizable nilmanifold symplectic form |
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